Boxes and Coins: Newsletter Challenge (06/12/07)

I think the practical limitations of a balance mechanism are mentioned somewhere , but I'm not sure at this late stage. A lot of Challenge Questions seem to hang on the issue of praticle vs theoretical. I'll leave you and Fyz to continue , but I can't resist :

.

uttered

Hi Fyz

My humble apologies for that last one (post 398). My error is so glaringly obvious, but sometimes I can't see the wood for the trees! 'Logic' was never a strong point with me.

Anyway, to give credit where 'tis due, the 20/40/... suggestion did not originate with me, but in post 100 (I think), long before I was born (into the world of CR4). With that I hope we can now give the coins and boxes a decent burial.

When I make some headway on the bouncing balls problem, I'd like to engage you on that, but it may take me several weeks.

Regards, =TeeSquare=

Hi TeeSquare

That's fine - I make plenty of mistakes on the way, but hopefully I usually get there.

On which topic - bouncing balls: if you are looking at the conditions for perfect elasticity, this is a difficult problem, and some of my thoughts turn out to be incorrect. My current view (still capable of modification) is that the mode structure of the pair of balls that are bouncing must in some sense be be 'compatible'. This could be balls with sets of modes that allow the wave to return to the originating point at the identical time; or in the case of solid balls, where the balls are so rigid that you can neglect propagating waves. For tennis-ball bouncing off basket ball, the tennis ball is quite rigid, and we rely on the wave propagating around the basket-ball. Solid balls usually rely on high rigidity. In practice, the difference from perfect elasticity is quite significant, but the observations for a single bounce are qualitatively similar.

Regards

Fyz

Hi Fyz

Following my visit to blunderland in post no 398, when I took another look at your proposal for selecting 11/33/55/77/99 coins each from boxes 1 to 5 and 6 to 10, it occurred to me that a further refinement is possible. Of course it must be assumed, as stated in the problem, that the scales are 'accurate', which I take to include any weights used for 'balancing', and uniformity among the coins themselves.

Instead of using a few 1.1 gm weights, we could do with just two different weights of 1.65 gm and 2.2 gm. Suppose the side with coins from boxes 1-2-3-4-5 is lighter, place the 1.65 gm weight on that side. If it goes down the fake coins are from box 1; if the scale balances they are from box 2; and if it remains up the second weight of 2.2 gm should be added. Now if it goes down the fakes are from box 3; if the scale balances they are from box 4; and if it remains up they are from box 5. A similar logic holds if the other side is lighter.

We would require at least four 1.1 gm weights and upto four 'placements' to exhaust all the possibilities, which could be interpreted as four 'weighings', though there would not be a 'balance' condition in any of them. With the two weights as suggested above, we could make do with just two placements, by exploiting the facility of up/down and neutral. How far the original problem conditions are met can remain a matter of semantic dispute.

I am a bit disappointed that there was no comment at all regarding my suggestion for using a straightedge scale as a balance. No doubt everyone has quite justifiably moved on to more interesting topics.

In retrospect, this is not the kind of problem I should have got into at all, but I have learnt some lessons from my involvement therein. I think the bouncing balls will keep my mind exercised for some time to come, but I'll have to find time to hunt up some references and do some reading first.

Regards, =TeeSquare=

Hi TeeSquare

Somewhere in this interminated (sic) thread*, I referred to the traditional method of weighing using a balance and weights. In that case, one weighing would comprise placing the objects to be weighed on the balance, and then adding and subtracting any number of calibrated weights until your have either achieved the best balance you can, or you have weights that are close enough together (for your purposes) that unbalance the scale to either side. You would use standard calibrated weights - the relevant ones presumably being 0.05, 0.1, 0.2, 0.5, 1, 2, and 5 (only one of each needed - though a typical set will have two each of the 0.2 and 2gm weights)

*Not worth searching for - plus I wasn't this explicit about what was involved.

Regards

Fyz

OK, I believe I have a "balanced" answer!

In the real world, this problem as given, has several real world solutions.

Before I go into detail, let me restate several things (see my post #160, if you wish), and maybe add some new input.
First, someone else has already examined the coins, and determined the box of counterfeits weighs 5 grams less than a box of good coins. This person and their knowledge of which box is not available. The only information left to us is the weight difference.
We are not given the weight of a good coin, but if needed, this information can be found by serching the Internet, or consulting a reference book, or calling the appropiate government agency or your local library. Some methods do not need this information.
The method used depends on the type of scale available.
Let's define a weighing as one READING of the WEIGHT indicated by a dial, digital readout, or count of reference weights on a balance scale.
Assume the weight of the boxes when empty is identical or any variations are so minor as to be irrelevant.

The best overall solution for its simplicity and elegance, is KennyT's radial balance. Place the boxes an equal distance apart on the disc and lift. The lightest box, holding the counterfeit coins will be at the highest point. No need to know the weight of a good coin! Also, no need to read the weight, as the only information used is that the box of counterfeits is lighter than a box of good coins. (Side note to KennyT: You've probalbly already thought of this for your production model, but to facilitate placement of items on the disc, have degree marks around the outer edge.)
I have to admit I had in my mind a different image of the construction of the radial balance. I imagined a conical pivot with a rounded point. Items to be weighed would be placed in a pan with a "dimple" that mates with the pivot point, and the pan is placed on the pivot. This way, you can have a set of pans of various diameters to accomodate items of different sizes.
However, as I said in post #160, this type of scale is not very common. In reality, we need pracitical solutions that meet the constraints of the challenge.

Now, let's consider a balancing scale with 2 pans, such as this one: http://www.scalesgalore.com/oharvdo.htm.
Place the boxes in a single line in order from #1 to #10 in front of the scale. Take a box from each end (first time will be #1 and #10) and place them on the plates, left end box on the left plate, right end box on the right plate. Repeat if needed until you have an imbalance. The box you just placed on the higher side is the box of counterfeits. Notice that no reading of weight is taken.

The next method is for a scale that has only one pan, plate or platform on which to place objects to be weighed. It does require that we know the weight of a good coin. Because we are handling the coins, we obviously know what kind they are, and we can access the needed information by the means noted above.
Before we put anything on the scale, we need to do a little math.
After finding the weight of a good coin, calculate the wieght of 945 coins.
Remove 1 coin from box #1, 2 from box #2, etc. Keep the coins from each box separate from the coins from the other boxes.
Place the boxes on the scale and take your reading.
Subtract the weight of 945 good coins from the reading and divide the result by .05 grams. The answer is the number of the box with the counterfeits.

These methods work with any coin in mass circulation. We can assume the coins in question are in mass circulation or else we would not have a thousand of them. Many collectible coins are no longer in circulation, and they require different methods of detection. With that said, we could just dispense with the scale and examine a coin from each box and look for signs of counterfeiting.

I have noticed on the Internet that some variations of this problem state that the counterfeits weigh more than the good coins. Methods 1 and 2 would still work, but look for the heaviest box. Method 3 works also, but the divisor is larger. Find the divisor the same way used in the given problem: divide the diffence between the weight of a good box and a bad box by the number of coins an individual box is to have.

I realize that Method 3 is close to the official answer. I believe I have clarified the official answer and made it adaptable to the real world.

I'm sure other solutions exist, or could be created. The ferris wheel mechanism is a variation of the radial balance, but in a vertical plane. Kris' sketch in post 191 looks intersting. Then we never fully explored the buoyancy possiblities in posts #243 - #246 and #252.

Now, to deal with a loose thread, so to speak. At one time I thought I had a solution based on approaching the challenge as a proportion problem. Well, it didn't work out. If someone wants to drive themselves nuts with that angle, go ahead!

Hi 3Doug

Surely method 3 only works as you describe if the boxes weigh nothing - otherwise you have to add the weight of ten boxes to your calculated weight. That was one of the two reasons that I regard it as daft, even if you knew the nominal weight of the coins.

The other was that, even if you knew the weight of the boxes, you would need the actual weight of the coins to be equal to their nominal weight to a precision of better than 25-mg in 945 coins. If the coins weigh even as little as 2.5-gm each (that is the weight of a U.S cent - one of the lightest coins you can find), the coin weight would have to be accurate to 0.001% - and that is with a perfect (rather than "very accurate") scale and with the measurement made under ideal conditions.

If you know the weight of the coins, it would be far less problematical to weigh the coins you have removed rather than the larger number left behind in your zero-weight boxes (yours, because I don't have any of those). Even then, the coins would still have to match their nominal weights to within 0.02%.

Weighing (measuring the weight) using 1,3,5,7,9 coins on each side and a traditional two-pan balance and the traditional method of adding calibrated weights would still mean that the average of the good coins would have to match to 0.08%.
That is more precise than would be expected for light (=low value) coins, even if they were clean and fresh from the mint. Even so, it would be more reasonable; it would also suggest that the problem originated when this was the standard method of weighing, and that the problem has been resurrected without a sanity check on the "official" solution.

Fyz

P.S. You show great charity in working so hard to try to support the official solution. Unfortunately, the words silk purse and sow's ear spring to mind.

Hi, Fyz

I was not trying to support the official solution. I saw the simplicity of weighing the boxes rather than trying to juggle the coins and keep them separated by the box they came from. I was trying to use that aspect to come up with a workable real world solution.

Considering the real world, when was the last time you saw coins in a box? I don't know how you handle them in the UK, but over here we often use rolls made of paper or plastic, unless we're talking about a large number of coins. Then the box will likely have rolls of coins.
I have a regular practice of saving my change and putting it into wrappers when I have enough to make a roll. The wrappers I use come in diefferent sizes to accomodate different denominations, and flattened out. The longest ones are 4 1/4 inches (10.8 cm) long when empty. I'm sure that if we use wrappers instead of boxes, we can ignore the weight of the wrappers.

If you insist we use boxes because the challenge says to use boxes, then what's to prevent us from calling the manufacturer of the boxes to find out how much they weight? Or using the Internet? Or taking the dimensions of the boxes, calculate the volume of mass of a box, and then calculate the weight of a box by using the weight per volume of the material used? If we can find the weight of a coin, we can find whatever information we need to calculate the weight of a box. Even if we do decide to go with wrappers, and their weight becomes an issue, we can calculate their weight too.

Hi Doug

So now we need to search for two bits of information, make assumptions about density and equality of the weights of boxes...

But my main point is that weight consistency for coins of 1 part in 100,000 does not form part of the "real world" I sometimes think I inhabit.

Fyz

[I doubt that it's at all relevant that I have wasted much time assessing measurement (of time sources) approaching 1 part in a thousand million million (=1E-15)]

OK, Fyz, here's a way to get around the box weight issue without have to look up additional information.
Stop thinking like a physicist, engineer or mathematician. As scary as it may sound, think like a lawyer instead.

The problem says you are limited to one weighing on the scale. That means YOU are allowed only one time to use the scale. It doesn't say anything about not having someone weigh something for you.

While you are looking up the standard weight of a good coin and making your intial calculation, have a co-worker weigh an empty box for you. Then calulate the weight of 10 empty boxes and add that to the weight of 945 good coins for the amount to subtract from the weighing you make. Or have the co-worker empty all 10 boxes, one at a time, and weigh the empty boxes and refill them..

After yor boss congratulates you in front of the entire staff, ask to speak to him in private. Tell him to stop being so cheap and buy a machine that will sort, count and wrap coins automatically, and it detects counterfeits too. Then point out he will not have to pay employees while they are dealing with these kinds of problems instead of taking care of customers. If he says he will fire you for insubordination, tell him you're obviously smart enough to go into competition with him and drive him out of business. You walk out of his office with a raise, a promotion, and a bonus. (OK, that last part is over the top, but you see my point.)

If KennyT gets to think creatively, so do I.

Hi Doug

I submitted your proposal to learned counsel. She said there was a remote possibly work in some legislatures, but:

In England, Lord Denning sitting in appeal had ruled that, unless you or the other person was the owner of the scale, there would be problems in getting permission to use it. This ruling had stood for about the last forty years, and now appears unlikely ever to be challenged.
I asked counsel if there was an out - and she said there wasn't anything about not using anything else, including another very accurate scale if you could get your hands on one. However, the coins are not allowed out of the building, and I'm not allowed to bring in any unauthorised equipments.

She is a capable lawyer and asked for background. Like Lord Denning, his first degree was in mathematics, so she told me not to be so silly, and that I should weigh the coins I had removed from the boxes instead.

She also asked ("just out of curiosity") why I thought I needed to weigh coins from 10 of the boxes, as you could get all the information you needed weighing the coins from 9 boxes.

I then consulted a business consultant, who told me to look into why I was only allowed a single weighing. It turns out that the bank for which I work is highly unionised, and that the reason that only one weighing is allowed is that it has to be performed by members of the appropriate trades' union, who are only trained to use the sort of two-pan balances that were the only precision machines up until the 1950s. Because each weighing requires a number of steps adding and removing the calibrated weights, they are only allowed to execute only a limited number of weighings in any week in order to minimise the risk of repetitive strain injury.
They have offered to retrain to use modern digital balances so they can perform more weighings if the bank will double their salaries. As there is just adequate capacity as things stand, we have impasse.
Regarding setting up in competition, the area of work is highly financially secure, and the vetting process is under the control of members of that same union.

Because of the security requirements, the bank has placed me on gardening leave for the coming year, and I will receive neither overtime nor my annual and Christmas bonuses, which together comprised 3/4 of my take-home pay. To add substantial injury to insult, both learned counsel and the business consultant have submitted jaw-dislocating invoices.

I suppose that's what comes of trying to think like a lawyer when I have neither the skills nor the background knowledge of the law.

Fyz

Well, once again we run into cultural differences! Bank employees in the US are not unionized, AFIK. The restriction on using the bank's equipment is either due to time constraints on that particular employee, or the manager's decisions. Generally speaking, if a worker needs to use a particular piece of company equipment to perform an assigned task, they can use that equipment unless it is reserved for use by certain personnel, and in that case exceptions will most likely be made.

Plus, lawyers in America are always looking for loopholes! That's why I came up with the approach I did.

That situation isn't real here either (though unions still have some influence). It's just my distorted sense of humour.

I happened to pass by the Temple, and expected it to be deserted or even derelict. But lo and behold, I find that the High Priests have resumed their deliberations on the profound mysteries of coin weights with renewed gusto. The prospect of any messiah cleansing the stalls of boxes and coin-changers in the near future appear dim indeed, though the posts have stretched into the fifth century since the altar was dedicated.

Considering that the topic refuses to die down despite sundry protestations regarding irrelevant content, let me venture into the hallowed precincts once more, and try to promote my 'simple' solution, which went unnoticed amidst all the verbiage in the latter half of the fourth century, when the attention span and concentration of readers must have been at a low ebb.

What I would claim for my method is that it meets all the semantic requirements of the original problem statement, such as one weighing (however interpreted) on an accurate balance (type not specified, but should not unduly stretch one's imagination). It uses the only bit of numerical data provided (one box weighs 5 grams less than each of the nine others). There are no questionable assumptions regarding individual coin or box weights either, regarding which the problem itself was vaguely presented. The procedure (already outlined in post 360 as guest) can be described in four steps as follows:

(1) Use a straightedge (say a wooden meter scale) as a balance beam by supporting or suspending it at a pivot point just above its centre of gravity so that it hangs level when unloaded, though sensitive to small loads of 'gram' magnitude;

(2) Mark five pairs of points symmetrically placed on either side of the pivot (they need not be equi-spaced, but each pair must be equidistant from the pivot, for which the scale graduations could be employed if accurate enough);

(3) Suspend the ten boxes from these marked points using suitably looped strings of negligible or equal weight, which will cause the scale to tilt one way or the other; and

(4) Find the position on the scale where the placement of a 5 gram weight brings the beam back to level. It should be right above the defective box.

That's about all. Certain practical considerations and numerical values had been elaborated in post 386, in case anyone has the patience to refer.

One could argue that in the process of hanging the weights one pair at a time, the light box will get detected anyway by causing unbalance (equivalent to 3Doug's two-pan procedure in post 405). But let us avoid any quibbling about multiple weighing, and assume that all the strings suspending the boxes are accurately positioned on the scale before making a clean lift to see which side is lighter, and then do the actual 'weighing' to achieve a balance condition.

I think this method is an improvement over KennyT's solution which does not use the information that weight deficit is known to be 5 grams. It would be interesting to know the experts' objections to my proposed method, and how it is rated on elegance.

I'll look in here again by and by to see if there are any developments. But meanwhile I am still pondering over the bouncing balls problem, which I consider far more worthy of serious technical discussion, though one can't help being reminded of that old limerick "... the one that was small was of no use at all but the other won several prizes". When I arrive at some idea worth sharing I'll post it in the appropriate place. No point in shooting off whatever comes to mind.

=TeeSquare=

It's an theoretically elegant valid solution, but the tolerances would have to be even tighter than I understand you to have indicated, because you are not just looking for the equilibrium at five grammes, but the differences between any of five possible equilibriums. I think the reason I missed it first time round was that I saw the words "wooden ruler" and "suspend the ten boxes" and moved on. BTW, I wouldn't be too worried about the ruler bending, because I'd suspend it on edge. What would worry me was the precision required of the positions. I haven't been through the sums in detail, but suppose we hang the boxes at +/- 0.6, 1.8, 3, 4.2 and 5.4 inches, you would be looking to for a maximum total error of +/- 0.6*5 = +/-3 gm-inches**. I think that would correspond to a placement matching for the boxes in the region of one-thousandth of an inch if the weight is to be closest to the relevant box.

KennyT's 2D solution also requires rather higher accuracy than I think that people have realised, but it would be a factor of about 3 times less critical than the linear scale (for the same maximum diameter). (i.e. 3 thousandths of an inch for a 12 inch diameter disc)

**Sorry about the weird mixed units...

Regards

Fyz

Hi, TeeSquare,

Love your analogy of the Temple and the moneychangers.

Wouldn't adding the 5 gram weight to the lighter side, no matter where it was placed on that side, bring the beam into balance? I'm not sure you can isolate the box of counterfeits this way. I don't know for sure at this point, so I guess I'll have to study up on it, unless you or Fyz care to enlighten me.

Hi Doug

A thought experiment: placing the weight straight above the pivot will obviously have no effect. If you it very close on either side of the pivot, it will have very little effect. The effect of the weight will increase as you move it further away.

As you move the weight away from the pivot, the effect will increase linearly - so it should only balance when it is over the offending box (i.e. it is in the same place as the lighter box - so it's just the same as putting it inside the box to make all the boxes the same weight).

If that is not already obvious, it is a case where a little experimentation is worth a thousand texts. I strongly recommend trying what would happen using a ruler*, a pencil, and two modest weights of different sizes. Place the pencil on a flat surface, an the ruler on top of it (with the pencil going across the middle of the ruler). Now put the lighter weight at one end of the ruler, and try what happens as you place the heavier weight at different positions on the other side. (Putting pairs of other equally-sized weights at equal distances on opposite sides will change the sensitivity, but not the optimum balance point)

Regards

Fyz

*Preferably not one that aquired its position through politics - you need one that is straight.

A ruler is a person who governs

Hi 3Doug,

Thanks for your remarks. I would say your doubt has been addressed rather lucidly and elegantly in the thought experiment by Fyz in post 416. (I would probably have gone about it in a convoluted way!) Fortunately the continued activity in the temple has resulted in my solution being at least read and acknowledged, so my maiden venture into blogland has not been altogether futile.

But Fyz has raised an interesting and valid point when he concludes by stating that adding more weights will '...change the sensitivity...'. This is something which can be discussed in another thread perhaps, as the whole business of sensitivity, accuracy and precision is an area where the words are often used loosely even by engineers who should know better (including myself).

Fyz has also pointed out the high accuracy required for the moment arm lengths. Very true. I can only use the excuse of "...very accurate scale..." mentioned in the original problem! In any case the problem itself and its official solution fairly bristled with practical and theoretical deficiencies (which is often a feature of real-life situations too).

I have made a check of what could be practically feasible with my method, and the following is a summary, without elaborating on the whys and hows:

<>Single coin weight can range from say 2g to 8g, whence each box with 100coins would be in the range of 200 to 800 grammes. The higher weight is more critical and the 'design' is essentially based on this upper limit. It is assumed that nine boxes along with their suspension strings weigh exactly the same (within 0.5g of each other) and the tenth is just 5g less.

<>An 'accurate' wooden foot ruler would be a better choice than the metre scale I had suggested originally, to be used edgewise to prevent excessive deflection. The boxes should be 'hanged' over the top edge at nominal distances of say 10cm, 11cm, 12cm, 13cm, and 14cm on each side of the centre. It must be assumed that the boxes are less than 2cm wide, and they need to be vertically staggered.

<>These moment arm distances on each side should be equal within about 0.01mm for each pair (tough!), in order that the probable cumulative unbalance due to such errors is noticeably less than the effect of a 5g unbalance at 10cm.

<>The top edge from which the boxes are suspended is in effect the line of their centres of gravity, so the midpoint of the scale top is the combined CG of all the weights when balanced. The pivot should be at a height of about 2mm above this midpoint for an adequate sensitivity of say 2 to 3 degrees of angular tilt when the weight deficit is 5g at any one box.

<>If the coin boxes are lighter, the accuracy requirements can be relaxed in proportion, and the pivot location should also be slightly higher to prevent excessive tilt.

I don't expect anyone (not even indefatigable Fyz) to check out all the foregoing numerical stuff. I've posted it only because I wished to round off the exercise for my own satisfaction. With that I hope the boxes and coins topic can now be left in peace.

Regards, =TeeSquare=

Hi TeeSquare

Some hopes that I would let it rest on that basis...

I may be misreading what you write, but if I interpret "is noticeably less than the effect of a 5g unbalance at 10cm" correctly, it appears that (at least) one of my points was not clearly presented. What we need is for the total "Error" moment (due to positional errors) to be less than half the difference between the closely-spaced "Signal" moment (due to moving the lighter box one position) - then we can discriminate between signals. In your case, the spacing is 10-mm, so the Signal is 50 gm-mm, and the Error needs to be less than 25 gm-mm. Taking your total weight on each side of 4-kg, the average positional mismatch** of the boxes needs to be 0.00625-mm.

If you space the boxes with equal separations of 32-mm, the allowable mismatch increases to a (marginally less improbable***) 0.02-mm.

**The mismatch is relative to the position of the pivot. As the error on the boxes averages statistically, inaccuracy in the pivot position will be the most critical. It needs to be less than half the values given, or ~ 3-microns and ~ 10-microns for the two cases.

***Most improbable for a wooden ruler. Some of the better metal rules can achieve the necessary 10-um (if you can still get them), but would need strengthening, and the alignment precision is only between marks on the same side, which would make suspension tricky.
However, on reflection, I think the 10-um constraint could be met using fine-threaded studding of the kind that is used for precision positioning stages

Fyz

Hi Fyz,

Must say you caught me out on that very promptly indeed -- can only admire your diligence. I discovered my folly about two hours after it was posted. It is quite evident that the difference in tilt angle between adjacent positions of the 5g weight will be indistinguishable from any tilt due to errors in the system, as the 1cm spacing between suspension points nullifies all my other claims regarding accuracy!

So I must throw up my hands in despair at not having arrived at any 'practical' solution outside of a metrology lab. At best I would go back to the metre scale and suspend the boxes at 10/20/30/40/50 cm on either side of the pivot, while demanding micron level accuracy in their positioning. KennyT's radial disc concept is admittedly more elegant, but there too any slight variation in angular or radial location of the weights can lead to ambiguity in identifying the culprit. Still, it may be easier in practice to locate ten holes very precisely around a centre, rather than along a line. I remember you had already made an assessment of the accuracy required.

Of course we all knew that the problem was a lousy one, but it has served to exercise my grey cells a bit, after ages of disuse! Now I wish to requiescat from all this cant, at least in semi-pace, hopefully at a safe distance from any boxes, particularly those which may be around 6' x 2' x 2'.

A little reading about bouncing balls has convinced me that the problem as posed is indeterminate, but I'm curious to know what additional assumption(s) could reasonably be made to predict the real-life response without gross error.

Regards, =TeeSquare=

Hi, Tee Square

I've been thinking about your device and I think with a few modifications it could work.

I first thought about using something more rigid that a ruler, such as a channel or square tube. That would reduce the sagging problem. But then the issue of spacing between boxes came up and required a long beam. This reintroduces the sagging issue.

Then I thought about adding a cantilever system to the top of the beam to deal with the sagging. But the cantilever would interfere with placing the 5 gram weight.

So, here's my latest concept: Take 3 equal length pieces of channel that are wide enough to hold the boxes, and flat strapping the same thickness and material as the channels. Cut the strapping into pieces that will fit between two channels when they are placed to form a "box". These are diaphragms that will reinforce and stiffen two channels to form the beam. Attach the diaphragms to the inside of one channel, equally spaced from the center. Place the second channel on top of the diaphragms and attach to the first channel. Then attach the third channel, open side up, to the top of the beam. If you wish you can mark the tray like a ruler with distances indicated from the center.

This 3rd channel serves as a tray to hold the boxes of coins and the 5 gram weight. The pivot point will be at the center of the length of the beam but above the horizontal center line of the middle channel.

If you have difficulties visualizing this, I'll be glad to draw it up for you. I'll probably do that anyway, because I need the practice.

Hi 3Doug

That took some thought - but still the main problem will be to achieve the required precision. If the length of the beam is 12", you would need to locate the pivot to an accuracy better than 10-um (0.0004") relative to the average position from which you suspend the boxes. I think that is quite difficult using any sort of assembly unless it is naturally self-aligning.

This was why I suggested using threaded rod (known over here as "studding" or "stud"). For reasonable quality threads, the manufacturing process tends to ensure better thread-to-thread tolerance than you might expect; and if you use 10-mm fine threaded stainless-steel studding (an imperial equivalent approx 0.4"/0.04" might be available in the US), that should easily support the required weight with a length of at least 1-metre - which would increase the allowable tolerance to 30-um (just over 0.001" in $US). You could then half wrap fishing line around the stud, both to suspend the stud and to suspend the boxes from the stud - if the two strands of the fyzhing-line converge away from the studding, that will bring the effective suspension point for the stud above the centre-line; similarly, the suspension point for the boxes will be below it.

The only difficulty that I can see (and I don't know if it is a real difficulty) is if the stud tends to twist out of the saddle of fishing line when you hang it* (because even that thread is a bit on the "open" side). If it turned out to be a real problem, I would bind two such studs and an unthreaded rod together, which would allow the outer edges of the two threads to be aligned (and the rod helps keep everything parallel).
*I did think about stabilising it with plasticine , but that would lose the tolerance.

Regards

Fyz

The problem with threaded rods is that they are bendable at the grades commonly sold. You would have to find rods manufactured to aviation grade, which would probably be expensive. Or buy common grade rods and have them hardened, which would also be expensive.

Noitice that I did not give specific dimensions for my rigid beam. We could detrmine the specifications as needed to do the job, and make them as precise as needed, or even set them to a higher precision to make this device useful for more than one specific job. With a 3D parametric modeling program, we can create a virtual model of the beam assembly, assign a material type to it, and let the program determine the center of gravity, which will give us our pivot point. Or we could figure it out ourselves.
Of course, when constructing the beam, precision cutting and placement of the diaphragms is paramount. We also need to ensure the channels are the same length and quality of material. We also need to be sure that our method of attaching the elements together will not alter the precision of the whole assembly.
This will take someone more knowledgable than I am to determine the specifications.

So far as I know, hardening does not significantly affect the basic rigidity (Young's modulus) of steel - only hardness and yield point. If we use a single rod of 0.4", the peak tensile stress will be less than 8,000 psi; I believe that typical yield points are in the region of 40,000 psi, so that is already a good safety factor. The total sag would be less than 0.2" at each end. (Short version: costs need not be increased for hardening)

(BTW, I didn't do the sums first - jut thought this diameter must be safe if my garden fork had survived being caught up in roots so often.)

If you bind two threaded rods and an unthreaded rod together as I proposed, you will see at least a factor of three improvement.

If you have a low-cost source that can work to tolerances of better than 0.001" over lengths of 3' or more, I would be interested to hear...

Regards

Fyz

Fyz, just how precise does thing need to be?

For the problem, as given, remember what I said in an earlier post:

"From the wording of the problem, we can infer that someone else has already examined and weighed the coins. But now that person either cannot remember which box has the counterfeits, or they quit, got fired, got sick, died, or maybe gone on vacation and can not be reached. It is now our job to identify the counterfeits, and we have to use the equipment at hand, and quickly too."

The weight of the bad box has already been found to be 5 grams less than the weight of a good box. The largest variation from an exact measurement would be + .1 gram for the entire box. The same variation would apply to a box of good coins. When you compare the maximum variation of one bad box against the maximum ariation of 9 good boxes, + .1 gram vs. + .9 grams, how likely is that going to happen? What is the maximum variation per coin? Could the accumulative effect of these variations affect the outcome?

If they do, then we need to worry about the precision of the mechanism. If not we can use my rigid beam according to method #2 given in post #405.

I know I should have done the math on this myself, but I don't deal much with significant figures on a daily basis and I really do need a review on them. Just one more thing to work on.

Now if we are using the scale to identify the bad coins in the first place, then we probably want as much as we can get. Or afford, providing we don't spend counterfeit money on it!

Hi Doug

My sums which gave the numbers in post #422 are based on the boxes having exactly equal weights, except for the fake box which is exactly 5-gm below the weight of the others.

Based on a 1-ft beam, that requires that the position of the pivot is accurate (relative to the average position of the hanging points) to better than 10-um, or 0.0004". With a 1-metre beam the allowable error would increase to 32-um, or 0.0012".

At the maximum error, the 5-gm weight would be half-way between the suspension positions of two of the boxes, so you couldn't tell which box contained the fakes.

Any mismatch in the weight of the good boxes would just make the position worse. A total mismatch of 0.5-gm* could be sufficient to invalidate the whole process, though it would more typically be tolerant of up to 1-gm - but that would require that the geometry of the balance was absolutely perfect.
*This depends on where the mismatch occurs - if more of it comes from near the pivot, we can tolerate more mismatch.

If we are to make a reliable measurement, we have to make the worst-case assumption, so a total mismatch of 0.25-gm would require that the positional errors be halved. It's all possible with top-quality well-maintained machining equipment, but you would need to use pairs of holes to align each suspension, and the work would need to be turned 180^O between the creation of the two sets of holes (that would correct for systematic error in the positions of the suspensions relative to the pivot suspension). I'm assuming here that there is no error in the attachment for the suspension - but this could be ensured by using fishing-line through the holes, as with the threaded rods.

Regards

Fyz

Hi 3Doug,

I never dreamt that my belated proposal of a linear scale as balance would have led you and Fyz into an elaborate discussion on precision manufacturing and material properties -- more to do with balance design, while the boxes and coins have taken a back seat. These 'practical' aspects of balance beam construction go far beyond the confines of the original problem, and probably belong in a separate thread in the mechanical forum.!

Though the topic is of interest to me as well, it won't be practical for me to get involved on the conversational scale prevalent in CR4, as I manage to look in here only briefly after somewhat extended intervals (when I dial-up to check for e-mail and delete the junk), whereas any reading or composing gets done off-line. So I'll have to remain an essentially dumb RequiesCAT. Nevertheless, since your post 421 was addressed to me, I'm only too glad to respond with whatever input I can provide, pertinent or im. But I'll stay clear of all the subsequent developments concerning fyzhing lines, threaded studs, etc.

Firstly, your description is good enough to convey the concept fairly well, if not all the minor details, so a sketch is hardly necessary for this discussion. I too would normally go about any design problem in a similar manner by making pencil sketches* and trying to visualise the implications. Every now and then some numerical checking is advisable to avoid ridiculously large or small dimensions.

I do appreciate the fact that you are willing to spend time on working out a design solution for an altogether artificial and contrived problem like the present one. Such creative effort is never wasted in my opinion. Anyway, let me go on to remark on a few aspects of your beam design.

Making a composite beam using channels and spacers is an accepted but costly engineering solution to many problems, which is resorted to when a standard section does not meet some constraint of strength or deflection or some other practical requirement. If you do a rough calculation with some tentatively assumed beam dimensions, you may find that the deflection at the tips is within say 3 mm, which I would consider acceptable in the present case as the combined centre of gravity of all the boxes would shift down only by a much smaller amount, and hence the sensitivity will not be noticeably affected.

PLACING a coin box accurately ON the beam would be tricky, because its centre of gravity is undefined. My idea was to use some suspension method (say a thread or flexible wire) so that the load point is more easily located, but there could be better alternatives.

To minimise deflection when using slender bars or flats, one could consider a very wide isosceles triangle with some bracings like a factory roof truss, but inverted. If suspension threads are used, a little thought will have to go into making them hang clear of the stiffening members over a reasonable tilt range. Such details can wait until the basic scheme is finalised.

The real problem, as Fyz has pointed out, is the need for locating the boxes very accurately (within about 0.01mm) which is ordinarily achievable only if precision components and facilities are considered. I had not visualised the problem in that context, hence my 'opting out'. I'm sure that if one puts in sufficient time and effort it would be possible to come out with a design which involves only common materials, components and facilities. But after more than 400 posts I guess everyone else in CR4 has just moved on to more interesting problems. And I am only a casual visitor, trying not to get addicted.

If you would really like me to work on this some more (to match the kind of effort you have been making) I can do so, but it will take me several days amidst other commitments, and I'll probably need to display a sketch along with the post. I can manage to draw and save something in MSPaint as a bmp or jpg or gif file, or using MSWord97 'draw' commands, but wouldn't know how to include it with the message (I'm near computer-illiterate).

I do hope you will complete your design exercise with sketches, dimensions and calculations, to your satisfaction, and that my remarks have been useful in a general way at least. In a situation like this (design with hardly any conceptual constraints) it is unlikely that any two engineers will follow a similar thought process, and there could be a whole lot of technically valid and elegant solutions, of which only a few (or none) meet the economic constraints! The present problem is however too peculiar to have anything more than curiosity value, so the designs will have to remain paper experiments, though certainly educative.

Regards, =TeeSquare=

* (Having become rather fossilised in the tee-square and slide-rule era, I decided not to use any software for drawing or other technical work, so my PC remains little more than a glorified typewriter. I wouldn't have admitted this in an engineering forum like CR4, had I not noticed the remark by Kris in post 14 of the thread on Tangent Circles that he never uses CAD! So I do have company.)

Me too - if it can't basically be conceived without tools, and primarily developed on the back of an envelope (sometimes several these days, as age sets in), I don't get involved, as others are much better at general assemblages than I. However, in the ends things have to be checked and communicated, and CAaAC (computer aided analysis and communications) comes into play. But I don't really use it for the basic design process - at most it's more a form of "virtual prototyping".

Fyz

Well, as I said in post #421, I would draw up my concept of the rigid beam mainly for the practice. Maybe someone out there will benefit from these illustrations.

The first two images are a profile (end view) and a lengthwise cross-section of my original concept. I included centerlines to indicate the approximate pivot point on the section view.

These next two are a simplified version of the concept, using only two channels.

As you can see, the pivot point placement on this questonable, being so close to the top of the bottom channel. Channels do come in different depths, so we can use a deeper channel for the bottom.

As Tee Square pointed out, placement of the items on the beam is critical. One reason I thought of a beam with the items on top, is that changing their postions is a matter of sliding them around. Having set points from which to hang items limits your ability to make adjustments. I was looking for something adjustable, and therefore reusable.

Fyz's threaded rod device might be a better solution, but I see one problem with it. Even though we call them threads, there is only one thread that spirals around the rod. Therefore, if you have a item with a length of string attached to its sides, and you hang the item from the threaded rod, the string will cross the rod at an angle, and this could affect the precision unless you are using an extremely fine thread.

A better approach would be to have a rod with precisely etched parallel grooves, or a smooth rod and use some kind of a clamp to hold the string in place.

I can now hear Pat Paulsen in the background saying, "Picky, picky, picky."

Hi Doug

Regarding the winding of the thread: so long as the supporting twine is symmetrical around the rod, the mid-point of the support will be the centre of the thread underneath (for the rod suspension) or above the (for the box suspensions) rod. For a 1-mm rod with 1-mm thread, the level of symmetry that you need is 3% of a turn, or 10-degrees. Aside from the tendency of asymmetry to cause the rod to rotate, this sort of error would be very visible if we hang a plumb line and look along the rod from one end.

My only concern about the method remains the possibility that the threading may cause the suspended rod will twist its way out of the suspension.

Regards

Fyz

Here's a way to suspend your rod to eliminate the twisting problem:

Drill a hole in a fender washer between the center hole and the outer edge. (A fender washer has a larger than normal OD.) Find the exact middle of the rod and mark it. Place a nut on the rod and run it down a short distance past the mark. Place the washer and then another nut on the rod. Run the second nut down close to the middle. Hold the washer over the mark and move one of the nuts up to it. Move the other nut up next to the washer also. Now, using a wrench on each nut, tighten both at the same time against the washer. This should lock the nuts against the washer and the rod. You can suspend the rod by passing your fyzhing string through the second hole in the washer.

This method is drawn from real world experience. Back when I sold auto parts, we often had customers bring in an old water pump for exchange with studs still in the hub. If the studs were tight, and they usually were, we would take two nuts and tighten them against each other. Then, using a wrench on the bottom nut, we could unscrew the stud. Having a washer between the nuts should not adversely affect this technique.

However, I can still hear Mr. Paulsen...

The risk here is that friction will hold washer slightly off centre - so the position of the suspension will no longer be determined by the thread on the rod. That was why I proposed to use multiple rods to reduce the longitudinal offset between the threads on the outside edges of the structure. However, as described, my original suggestion would have failed - the rods need to be touching at their high-points, not intermeshed. This doesn't need to be exact, so could be done by supergluing (as separators) short sections of wire that fit partway into the thread at strategic intervals.

Note that it is not critical that the fishing line used to suspend the rods is exactly at the centre of the rod - we can adjust the balance. However, it is critical that the hanging lines for the boxes are spaced the appropriate number of turns from the rod suspension.

Fyzzi, fyzzi, fyzzi?

Fyz

BTW, an ex-colleague of mine spent virtually his entire career developing and improving "ruling engines" to generate increasingly accurate diffraction gratings. But you're looking at sub-microinch accuracies there. Shortly after he retired, an amateur published a design for an engine that was nearly as good... http://www.britastro.org/iandi/manning2.htm

Well, if the washer is not centered, then the rod is unbalanced, and you can file the low end of the rod to bring it into balance, or add a thin nut to the high end and screw it on until the rod balances.

And Mr. Paulsen is getting louder.

The problem with the washer is not the unbalance of the rod due to displacement of the suspension (that can be corrected as you say, or with a bit of plasticine), but the displacement relative to the positions of the suspensions for the boxes.

Put a gag on Mr Paulsen: at this level of tolerance you just have to be 'picky' enough if the results are to be anything other than random.

Fizhy fizhy fizhy

Hi 3Doug and Fyz,

I'm amazed to find both of you still slugging it out over the intricate design details of a beam balance. The discussion is certainly interesting, despite the nuances of references like Pat Paulsen and other 'asides' going over my head (for obvious cultural reasons), though such diversions may be a key factor in the sustained contributions and enthusiasm of many regular CR4 members.

I'm naturally hesitant about getting further involved in this exchange for certain practical reasons (mainly my irregular visits to CR4). I may find I'm fyzhing in troubled waters! Anyhow, since I am making these remarks, let me also update you on my views as of now, hoping it won't be viewed as mere criticism.

Since my background has been in 'heavy' engineering, I would consider the use of 4" channels (as shown in post 429) or any such steel structural member to be inappropriate for a task like the present one where accuracy and lightness are desirable. Normally a 100mm channel could be used to support a load of the order of a tonne rather than a few kg. The standard rolled sections have rather coarse dimensional tolerances which make them economical for use in steel structures, but rendering them too crude for small or precise assemblies. A better alternative may be one of the light-gauge metal sections in steel or aluminium, avoiding any welding which leads to distortion. I also feel that intermediate ribs or spacers are superfluous, besides complicating the manufacture. Whatever we choose should have a section shape and size just sufficient to take the loads without excessive stress or deflection.

My experience with getting fine threads or precision components manufactured have not always been very happy. Consequently I try to visualise my designs as far as possible in terms of standard available components which can be cut to size and bolted together without using costly machines or special skills.

As the originator of the 'linear scale' balance concept, it is perhaps appropriate for me to outline how I would go about the design now (at least in hindsight). Describing it in words is rather difficult, but I don't know how to get a sketch into a CR4 post. Essentially I would use two thin steel flats with spacer bushes at intervals, forming an "H" about 30mm wide and 30mm high, assembled using small bolts, say M3 size. Then I would employ five stiff wires (say ~1mm) of varying lengths with their ends bent down to form hooks as suspension points, and some means for axial adjustment with respect to the centre. They will need some offset along much of their length so that they can run side by side, so that the hook points remain along the centre line.

I think the above arrangement with linked hooks is simpler than having a separate adjustment device at each suspension point.

The pivot should preferably be a knife-edge arrangement with a provision for vertical adjustment, so that the sensitivity can be tuned. A suspension pivot is also feasible, but it would allow the whole contraption to swing around and hence slow down the elaborate 'calibration' procedure which I have foreseen. Essentially, each symmetrical pair of suspension points should be made 'equidistant' from the pivot in a sequential manner using equal weights (exact distance being immaterial). The ends of the beam can have axial threads for balancing screws, to be adjusted during calibration. More details will be clear only if I can include some sketches and expand my description.

I think a one-foot beam would be too short for comfortable suspension and adjustment of ten weights, and would rather keep the nominal spacing in steps of about 8cm on either side of the centre, so that the whole beam assembly fits within a length of 1m. It should be reasonable to assume that we are dealing with weights which are accurate within 0.1 gram, considering that we are trying to detect an 'error' of 5g.

I hope that provides sufficient indication of my thinking. Now if anybody has any comments....

Regards, =TeeSquare=

Hi TeeSquare

What I still haven't grasped is how you intend to achieve the required symmetry in the positions of the pairs of coin-box suspension points around the suspension for the beam.

Regarding fine threads: - I agree that having individual rods threaded is probably a route to imprecision. Using "volume" studding works rather better, because this is a standard process, and errors tend to be very similar between rods of the same manufacture. So I have chosen rods of a diameter where a 1-mm thread is rather standard. If the threaded rods are placed in reverse direction (relative to their manufacture), any consistent errors should cancel; and it is quite easy with threads to see if there are mismatches (by placing them against each other and holding them in front of a uniform white background).

Regards

Fyz

Hi Fyz,

My fault I guess, for plunging into a description of constructional details without adequately explaining the underlying concept. Let me try to make amends.

Before that, let me say that I prefer not to struggle with problems relating to the accuracy of threads or any other matter involving high precision manufacture, mainly because inspection is a costly business which can often be justified only for mass produced items. My experience has been that inspection may be done using instruments which may be defective, or not properly calibrated, and I don't have the means to check it out or justify my doubts. Simpler to avoid precision requirements if at all possible. So I won't even try to comment on the design which you and Doug have been discussing.

To repeat, the problem I have visualised is to identify an object weighing W-5 grams which is visually indistinguishable from a group of nine similar objects weighing W grams each, using any 'mechanical' weighing device or contrivance, in one single weighing operation.

The value of W could typically be in the range of 200 to 500 grams, in the spirit of the original "Boxes and Coins' problem statement. I have assumed that we are talking of weights which are accurate within about +/- 0.1 gram. We are considering total weights of the order of a few kg at most. A similar problem in the range of grams or tonnes would invariably have triggered some different conceptual approach from the beginning itself.

The basic idea in the linear balance solution is to suspend the ten weights from defined positions along the length of a beam balance which is suspended or pivoted at the centre. If all ten weights were equal the equilibrium position of the beam should be horizontal. If one of the weights is deficient, then it can be detected by finding WHERE a weight (of magnitude equal to the KNOWN weight deficiency) should be placed for making the beam level again.

The practical concept in the design outlined in my post 435 consists of treating the beam as a cradle which holds five 'rods' or wires of different lengths (in my case 16, 32, 48, 64, and 80 cm, for a cradle about 90 cm long). The ends of these rods carry 'hooks' or suspension points from which the weights can be hung. The DISTANCE BETWEEN the hooks on each rod need not be a precise known value, as long as it remains INVARIABLE. The rods themselves can be individually positioned axially along the cradle by any suitable means (not described here, but could be a fine screw device). The total movement required for each rod will be less than 1 mm. Coarse and fine axial screws at the ends of the cradle can be used to compensate for any unbalance introduced by auxiliary items mounted on it, so that in the unloaded condition this assembly will be level when supported at the central pivot point. If it is a case of unstable equilibrium, fix a couple of equal dead weights below the ends so that the combined centre of gravity gets lowered sufficiently.

The calibration procedure is necessarily a bit elaborate. It requires two EQUAL calibration weights of about 5W grams each, to be suspended initially from the shortest rod, and the others in succession, in order to achieve the 'equal-distance' condition. The exact weights are immaterial as long as they are equal within say 0.1 gram, and they could even be a pair of sand boxes with a loop of thread for hook suspension. The choice of 5W is to keep the combined centre of gravity of the beam assembly together with the weights at the same level during calibration and the final weighing.

The first step is to adjust the sensitivity of the balance which should be 'tuned' to suit the total weight to be carried (typically 2 to 5 kg). This requires that the actual pivot point (knife-edge seat in my case) can be adjusted vertically with respect to the cradle. How this is achieved is a matter of mechanical design detailing which need not concern us here. The 'desirable' position of the pivot to get deflections within a reasonable range may vary from about 2 to 5 mm above the common CG, for the range of loads being considered. The sensitivity adjustment is combined with the positioning of the first rod as follows.

In the initial position the two weights will not be equidistant from the centre, so the rod will need adjustment to make it level. Since the slight movement of the rod might have upset the original balance, the weights should be removed and the empty beam levelled once more. Then the weights can be replaced and also interchanged to ensure that the beam remains level. By this process the two weights (inner rod suspension points) are made equidistant from the pivot. The sensitivity setting is described below.

A small pointer at one end of the cradle moves against a fixed blank scale on which a zero mark can be made for the level position. With the two calibration weights on the short rod (or indeed any other) the system achieves its final centre of gravity, so the pivot position can be adjusted once for all (provided all the hook points are at the same height). For sensitivity tuning, a 5 gram weight is placed at the 8 cm offset position on one side, and the corresponding deflection should be around 2 degrees (~15 mm at the end). The pivot should be adjusted to achieve this value (approximately), and corresponding marks can be made above and below the zero on the vertical scale with the weight on either side. If the 5 gram weight is moved to the 16, 24, 32 and 40 cm offset positions along the cradle on each side, the deflections should increase by equal amounts, and corresponding marks can be made (tentatively) on the scale.

During the pivot adjustment procedure, the effect of any errors in the weighing system itself such as friction at the pivot should be relatively small. Adding or removing a 5 gram weight should result in a demonstrably consistent deflection, if not exact. Each marking on the scale will in effect be a small band, maybe about 2 or 3 mm wide, which should be sufficiently far removed from the one above or below it.

Once the inner rod has been 'equalised', the calibration weights can be hung at the ends of the next rod, whose axial position should be adjusted and any necessary correction made for rod shifting, and so on for all the others, without any further change in pivot setting.

At each stage the end screws may need adjustment for balancing the empty beam, and interchanging the weights should not affect the state of balance. The 5 gram weight check at all positions should be made with the weights hanging from each adjusted rod, to ensure that the scale markings remain valid. If noticeable discrepancy is observed and the markings run into each other, it would mean that the suspension points are at different levels with respect to the cradle, or else the deflection due to the weights is excessive and a stiffer cradle is required. In both cases the CG of the loaded system is different from what was assumed for the sensitivity adjustment. Other errors such as loose assembly at the pivot can also cause erratic behaviour.

For creating the two equal weights in the first place (if they have not been weighed separately), the balance itself could be used judiciously by hanging the sand boxes from the hooks of the longest rod, and adjusting the contents until the beam tilt is unaffected when their positions are interchanged.

Once the calibration is achieved, the weighing procedure for the ten objects should be obvious.

I hope I don't need to elaborate any more on the concept. As far as the design is concerned, I have further refined my ideas and am now in favour of a pivoted hook with a leverage so that the axial adjustment for equalising the distances (through a few microns) can be done more easily. The overall length could also be reduced somewhat. You will note that a lot of the choices depend on assuming certain realistic numerical values for various parameters. If these values were significantly different some of the concepts may need to be changed in order to keep the design 'practical'. For instance I wouldn't consider a beam length greater than 1 metre. Let me not burden you with any more wild ideas! There is perhaps no end to the range of solutions which can be conceived even for a ridiculously artificial problem like the present one!

Regards, =TeeSquare=

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