Squares squared: March 2021 Challenge Question

Your graphics win the prize. Clear, easy to follow, succinct.

If only legislation could be so clear. Oh, wait a minute, that's not the goal.

The puzzle does not specify the four dots must be at the square vertices. A twelfth square centered with the whole pattern and 3√2 side length cross the four outer dots. A thirteenth square with 4 side lengths also cross the four outer dots.

If the segments can touch more than four dots then several additional squares exist. If these dots are not considered to be points and the lines are infinitely thin then an infinite number of squares exist.

Correction: The twelfth square has sides of 2√2 length.

True, it could be argued that if the "dots" are jumbo-sized as shown and not zero-dimensional, an infinite number of squares could be drawn using any four dots arranged in a square.

I don't know if the puzzle poser is that diabolical, although he or she could be.

Vertices VS Sides, Touch VS Penetrate

Redfred,

If you do not require that the dots are at vertices then any line crossing 4 dots counts as a genre of solutions. And any line crossing 3 dots plus another line crossing 1 dot counts. Each of these generates infinite numbers of squares. You did not carry your liberal interpretation of the locations of the dots on the square perimeter to their full consequences.

If you do not consider the dots to be points then you have to decide whether only tangents count(which "touch" the "edge" of each involved dot) or if you allow secants. That is, can the lines pass through some of the dot or is the dot a black hole destructively consuming any line which passes through a dot's "event horizon." If secants count then do they count as one touch or as two ? If secants count as two "touches" then a dot at vertex might also count as two touches and a dramatically different set of diagrams results where it becomes even more difficult to avoid excessive "touches." One interesting aspect of the secant vs tangent distinction is whether a single dot with its inscribed square counts as a genre of solution. Allowing rotation of that square gets an infinite set of solutions out of a a single dot.

______________

thewildotter

The litigious often cannot perceive the full potential scope of their own relaxation of definition. I am certain that someone can, with additional missing definition interpretation, find even more potential solutions than I have outlined here. For example, are the edges of the dots illustrated genuinely circles ? If you say yes, then take a magnifying glass and look closely at your monitor.

As a great philosophical question: "Do any real things actually exist in less than three dimensions ?" Please do not consider greater than three dimensions in the answer... Too far afield.

The dots are not defined as points. The connection between each of the four selected dots is not defined as a single line. Conceivably therefore, four selected dots could be connected with closely spaced parallel lines arranged to encase various dots within a small square (lines of the square tangential to the circumference of the dot). Further, since the connection between each of the four selected dots is defined as neither a single line, nor a straight line, double lines could follow a serpentine path between four selected dots, straightening where they meet each dot to again similarly encase various dots within a small square. Thus, chains and branching networks of double-line connections between various four selected dots could form an infinite number of dot-encasing squares. Three dimensions? Maybe the drawing is a top view of a cone or four sided pyramid, of indeterminate height, with dots on the side or sides (oblong individual dot geometry appears circular in the top view). Height being a variable - infinite possible squares.

... or more simply, as per three dimensions, it is a top view of 12 round pegs protruding from a surface. This thought reminds me of the 3-dimensional tic-tac-toe we played as kids - a transparent structure - four markers in line was a win - a line being horizontal, vertical, or diagonal.

We all know that assuming can easily lead to misconceptions on many levels, but in this case we MUST make some assumptions:

My assumptions for this challenge are two: A) Each dot MUST represent a single point, and B) These dots represent a perfect array, with the horizontal and vertical separations between adjacent dots all having the same value. A corollary to these would be that all lines joining adjacent dots are either perfectly horizontal or perfectly vertical.

If either of these assumptions is removed, then there can be no definitive answer other than zero or infinity.

Yowl !

It was easier to put the first cat in the bag than it will be to add the second cat. You seem to be intent upon adding a whole clowder of feral cats to this bag. It started out logically...

"The dots are not defined as points." I believe I asserted this concept with the discussion of tangents and secants. I further reinforced the perception by suggesting that the reader look at the problem statement representation of a dot with a magnifying glass.

But then it went bonkers...

"The connection between each of the four selected dots is not defined as a single line." Like tying the tails of four feral cats together before attempting to put them into the bag.

".. follow a serpentine path between our selected dots.." Yep, more like herding cats than defining a square.

Interesting that your set of lines are defining substructures of circumscribed dots. I circumscribed a single square with a dot. You are circumscribing four dots with four substructure squares to construct a mess with included squares as an example of a square which is similar to an heirarchical dyslexic wandering through your hierarchy wondering if there is a doG.

All that being said, it merely shows that absent definitions of terms, puzzles such as this one are likely to take off into the bizarre. I am (in full disclosure now) feigning objection to your dramatic proof of the point I was making. "I am certain that someone can, with additional missing definition interpretation, find even more potential solutions than I have outlined here."

Thanks for nailing it down so colorfully with your imaginative definition of a square as anything with four dots possibly each surrounded with a square. Now Schrodinger, I wonder if each of the cats in the boxes in this bag are alive or dead.

_____________

thewildotter

No squares can be created because the dot pattern is actually as shown in Diagram #1. The large flat panel on which the dots are painted has been set at an angle as shown in Diagram #2 to create the illusion of dots on a right-angled grid.

Now you are literally stretching it!

I took a screen dump of the original image, and pasted it into my CAD program. I then created an array of 0.375" diameter circles with a 1.50" horizontal and vertical spacing (Blue). I scaled the bitmap ever so slightly (1.500/1.458) to match it with the circle array:

I then added a black line from the upper leftmost circle center to the upper rightmost circle center, and a blue line from the uppermost left circle center to the lowermost left circle center. Finally, I added a red linefrom my best estimate of the center of the uppermost left dot to my best estimate of the center of the lowermost left dot.

That red line is barely visible in the full image, so here I include enlargements of those two dots and the line ends. The blue line is vertical, while the red line is about 0.5° off of vertical.

I don't see the keystone effect that you show.

You wrote in #13, "in this case we must make some assumptions", one of the assumptions being that, "these dots represent a perfect array". I agree that the dots must be perceived as a perfect array, but in reality might not be. As Redfred wrote in #10, "Many times the surprising key to a puzzle is found in the ambiguity of the stated conditions". Must we assume that the dots are arrayed on flat surface? If we assume the flat surface, can we further assume that the provided image is a view at right angles to the flat surface?

I don't have CAD on my computer. I sense that my sketch inadequately conveys my idea. Possibly you could in your CAD program, create an array of 12 dots on a plane such that the array when viewed from an angle appears to be perfect. That is the view, I submit, that the puzzle maker could be presenting to us. Then rotate the plane so that the dot array is viewed straight-on, and the actual not-perfect array will be displayed - the not-perfect array that the clever puzzle maker illusionist wants to conceal from us.

We don't have to assume a flat surface; the OP said: "You may want to print this one out!", Thus it will automatically be a flat surface (within the limitations of flatness for the paper or other medium on which it is printed).

In my #12 and #17 posts I do assume a flat surface, but I postulate that the puzzle presentation could be a view of a flat surface that is at an angle to us. If this is the case, note that in the postulated actual dot array when viewed straight-on (the one which the clever puzzle-maker wants us to be unaware) each dot will be an ovoid. This doesn't show up well in the #1 diagram in my #12 post (a rough representation of a straight-on view of the actual dot array). Of course, viewed at the angle presented to us (my #2 diagram) the ovoids are foreshortened and appear as perfect circle dots, and the dot array appears to correspond to a perfect right-angled grid. All visual sleight-of-hand! The puzzle-maker has only shown the dot array viewed from an angle, and has omitted from the field, a view of the edges of the angled flat panel (my #2 diagram), on which the dots are painted - to do so would have given his game away. These deceptive tricks can't fool me!

Regarding your second paragraph: Yes, I could mathematically produce the effect you are referring to in CAD, but unless the original challenge is changed, I won't bother.

What needs to be changed in the wording of the original challenge? I have presented a solution (no square possible by connecting four dots) that does not violate any terms of the challenge, and IMHO is more imaginative than most of the other proposed solutions. You took the time with CAD in a misguided attempt to discredit my answer (admittedly, my sketch lacks clarity), but now that you apparently understand that answer, it would seem that, in a more generous spirit, you would graciously take the time with CAD to support my solution and render it visually intelligible to all.

I have no complaint about the clarity of your illustrations. They do convey the concept you are describing.

I wasn't paying attention to the time, but I probably spent over an hour making that drawing in CAD, and that required no calculations. It would likely take me several hours to figure out how to create a mathematically correct illustration of your concept.

I'm sticking with Rixter's illustrations, that the dots represent points on a uniform grid/array, and a view looking normal (mathematical term) to the center of the image.

I understand. Thankyou. All of my posts have been in the spirit of play. It is, after all, a recreational puzzle. Where is the fun in being deadly serious about such a trifle? No personal injury or property damage is going to result from getting it wrong. I'm not suggesting we abandon logic, but it is more entertaining if proposed solutions also contain inspired nonsense.

Good point. I'm definitely more of a scientist than entertainer... (or entertainee).

How about bending it a little....The suggestion was made about printing the image. If I do so and cut around it (as in Rixter' picture #1), I can form it into an open top cube......one more square formed at top of the box. For what's left of my sanity, I'll not be exploring that train of thought further.

Nice!

I think the suggestion "you might need to print it" is indeed key to the interpretation.

Let us take it one step further and print it... on rubber! And then fix it at the edges, and put a weight somewhere, so that it is stretched like an interpretation of a gravitational field. Let's see where we can make squares...

As I mentioned somewhere else: if one wants to have exact mathematical answers, you have to pose the question in an exact mathematical way. But seeing the wide range of interpretations and creative answers is much more fun, and we learn much more from that, so in the end I am advocate of vague challenge questions :-)

Sorry, I have to go with "infinite", with one example above :-)

Nope! those lines are not "connecting the dots".

But the vertices of a square are only part of the square.

Minimal Unique Square Definition

An important concept to consider is that two diagonal vertices are sufficient to uniquely define a square. Two non-diagonal vertices will define two squares(a square and its mirror about the line formed by the two vertices.) Points along the edges of squares or points in the internal area are likely to describe an infinite family of possible squares. Alternatively, squares can be uniquely defined by a vertex and two other points given to be on the perimeter of the square if at least three sides can claim residence of at least one perimeter point. Note that this does not hold for points on line extensions beyond the vertices of the square which are not "on the perimeter."

Three guaranteed perimeter points may define zero (eg., some obtuse triangles), one (eg.. some right triangles), or greater than one squares (eg., some isosceles triangles). As an exercise fully analyze the square counts for various isosceles triangles with all the points somewhere on the square's perimeter...

___________

thewildotter

Note that the points here are mathematical, zero extent points not "dots." Also,, do not include any imaginary number or quaternion or octonion analysis and no 3D analysis unless you just like pain.

Oops ! All right triangles generate an infinite count of squares since the lines emanating from the right angle need not limit the length of the two sides they are defining. I got distracted because I was assigning the points to the two sides opposite the right angle rather than to the sides adjacent to the right angle. They could be construed either way and that means an infinite number of squares result.

____________

thewildotter

Sure look like the dots are connected. If the four blue dots were to represent villages, and the red line were to represent a road, a motorist would conclude that the villages are connected.

Thanks!

Such a question invites to a mathematical approach, but it has an incomplete mathematical wording - too much space for interpretation. It needs either exact mathematical phrasing or a physical (or other) context.

What if the question was along the line of: an autonomous vehicle needs to make deliveries, but the programmer / planner loves squares. How many routes could he create to satisfy his OCD? :-)

One remark: If I state that such a question needs a complete/exact mathematical wording, that should probably result in a single outcome with which everyone can agree. It is more fun, and everyone can learn more, if there is room for different views and interpretations of the question :-)

Wow. Do engineers have a tendency to go into the weeds?

No danger of going into the weeds if you stay on the road (the red line) referenced in my #25 post.

Taking the interpretation that each square must have a dot at each corner, I'd do this; Count squares from each dot in turn. For every square, my tally would include 3 that are identical squares. Being short on Typex, I'll run with that and give the answer as 1/4 of total derived.

Typex not needed. The outer 8 dots share a symetry. Each can be part of 3 squares, giving 24. Doing similar with the inner four dots, each can be part of 5 squares. That brings total too 44. For reasons of replication mentioned, the total of unique squares will be 1/4 of that. ie 11.

Here's a thought for anybody feeling a bit mad; For any regular grid of n by n dots, there is likely a pattern to progression of how many squares could be formed. If such a pattern could be deduced, it would be relatively trivial to apply it to 4 by 4 then deduct the squares formed by additional corner dots in the given picture (which I would make to be 9 in this instance). If it helps any, this occurs to me from visuals of Pythagoras theorm that use a square tilted within another. That may give some hint as to how to generalize a formula for n by n. Had the question involved, say, 9 by 9 dots (but with each corner one then omitted), such an approach might be the only way. I'll leave the thought there and go back to my cozy room with padded wallpaper.

The general sequence for an array with the corner missing seems to be 0, 1, 11, 37, 88

The following calculations start by temporarily adding the missing corner. As any array increases, more non perpendicular squares are added as 1 of root 2, 2 of root 5, 3 of root 10 etc. 4 of root 17 etc. A bit of cross-multiplication is needed to tally them.

Column and row labels indicate squares of given length.

1 2 3 4 5 ∑ correction ∑ Angled Squares Total

1 1 1 -1 0 0 0

2 4 1 5 -5 0 1 1

3 9 4 1 14 -9 5 4+2 11

4 16 9 4 1 30 -13 17 9+8+3 37

5 25 16 9 4 1 55 -17 38 16+18+12+4 88

That's likely as clear as mud. Easy enough to continue the process. Anybody got software or patience to validate the last two in the series?

Sorry all - format came out rubbish.

That's OK.

It's April. An official "answer" has yet to be posted, too.

Just as well the format came out rubbish - so was my logic.

No matter, it's still providing me with fun and a bit of brain stretching. If memory serves, there are a few others recently with no 'official answer'. That's unfortunate because the given answer often sparks good debate. This one is almost certainly solved, but the generalized case (a square array with corners missing) seems interesting.

Corrected version. Commas used to keep spacing.

,,,,1,,,2,,,3,,,4,,,∑,,,,,Cor,,,,,∑,,,,,,A1,,,,,,,,A2,,,,,,,,,A3,,,,,,,,A4,,,,,,,,,,∑

1,,,1,,,,,,,,,,,,,,,,,,1,,,,,1,,,,,,,0,,,,,(1x0),,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,0

2,,,,4,,,1,,,,,,,,,,,,5,,,,,,5,,,,,,,0,,,,,(4x0),,,,,(1x1),,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,1

3,,,,9,,,4,,,1,,,,,,,,14,,,,9,,,,,,,5,,,,,(9x0),,,,,(4x1),,,,,(1x2),,,,,,,,,,,,,,,,,,11

4,,,,16,,9,,,4,,,1,,,30,,,,13,,,,,7,,,,,(16x0),,,,(9x1),,,,,(4x2),,,,(1x3),,,,,,37

Slightly twisted way of solving by adding to the orignal diagram. Row 4 is included for anbody who fancies checking by diagram if I'm correct. Should be reasonably clear what line 5 would be. Probably not too hard to do a bit extra and generate a formula from the above table.

Ooop...7 on last row should be 17.

This is really pushing it - The suggestion is made to print the diagram. If that was done, a bit of judicious folding could leave any 4 dots in a square arrangement. That would give an answer of factorial 12. Well, some sort of madness like that.

Not factorial 12, but infinite. The folding could array any selected 4 dots into an infinite number of different sized squares. Similarly if the dots were printed or marked on an elastic membrane. Secure the perimeter of the membrane in a horizontal plane then pull vertically from the centre point of the array. By varying the tension of the pull the four dots closest to the centre will move through and be on the corners of an infinite number of different sized squares.

I knew it was wrong somewhere, but couldn't see the math. In short, any 4 dots could be squashed about to form a square. It't gets even more complicated by folding the print back on itself. My current problem is seeing if there is a finite answer. Dang the bloggers for not giving an 'answer'. I suspect that 'infinite' is wrong, but far be it from me to drag this out until we hear an official answer

As others (and I) have pointed out, the Challenge Question is not defined with sufficient rigour, and this has led to various proposed solutions which would enable an infinite number of squares to be formed. Those speculative solutions have been entertaining, but almost certainly the intent of the Question, and the official answer, is a finite number of squares.

I disagree. This is the type of loosely defined question that shows the creativity of both the individual answering the question and the individual presenting the question. This is an excellent employment question. With multiple approaches to achieve infinity for the total number of squares, it is informative to see which infinite approach is presented or the complexity of a finite answer is presented.

Alas at more than half way through April we still have no insight into the presenter's mind.

If the loose definition is to deliberately enable multiple solutions that would generate infinite squares, maybe the official answer, as per Georg Cantor, will be the solution that would generate the largest infinity of squares.

The multiple possible answers is precisely why this is a good employment question. If the job position is to be an Uber driver then the candidate answering "eleven squares" maybe the best choice. If the job involves engineering code compliance or composition then the infinite answer due to the dots size or not necessarily placing the dots at the vertex may be the preferred choice. Folding the paper may indicate sales is where a candidate should be placed.

What are your job placement recommendations for candidates who provide the given various answers to the following question:

How could you find the height of a tall building using a precise barometer?

Answers:

1) Drop the barometer from the top of the building and measure the time it takes to hit the ground. Use the distance equation for a falling body to determine the height of the building.

2) At a specific time of day measure the length of the barometer's shadow and compare it to the height of the barometer. Apply that ratio to the measured length of the shadow of the building to determine the height of the building.

3) Go to the owner of the building and say, "I have a real nice barometer and you can have it if you tell me the height of your building."

Well for the first two I'd start by asking how the precision of the barometer matters?

1. Physicist.
1. They naturally measure things by smashing something.
2. I would also ask them since you had to climb the building, did you consider counting the number of steps?
2. Surveyor.
1. I'd also ask if they accounted for terrain differences
3. Purchasing
1. How did you know they were the owner and not a doorman? Still, Doorman probably knows the height of his building.

We are in close agreement.

1) Task co-ordination/scheduling

2) Spacial visualization/3D rendering

3) Presentation of proposals, entertaining clients

The following I read as atributed to Elon Musk; 'I walk 1 mile south, 1 mile east, 1 mile north and am at starting point. Where am I ?'.

Just over a mile (about 1.16) from the south pole, and with an infinity of positions. The 1 mile eastward walk can led back to where it started. Obviously a replay on the more well known north pole problem.

A loosely defined problem does have advantages in drawing out different thought process. As such, there is no absolute 'correct' answer. How a person interprets a problem can be just as interesting as any answer.

You mean the north pole.

Nope. Consider a great circle of length 1 mile around the south pole. Start off 1 mile (anywhere) north of that.

There is no such thing as a "great circle of length 1 mile" anywhere on Earth. A great circle on Earth always has a length equal to the circumference of the Earth (considering the Earth as if it were spherical, which of course it is not).

...Or does the term "great circle" have a different meaning on your side of the pond?

Reword that to something like "a circle of 1 mile circumference centered on the Earth's axis near the South Pole", and I'll agree with you.

Who said I was on planet Earth, generally people think I'm not ?

OK, ya got me ! The correct word escaped me (and alarmingly it still does). 'Latitude' seems a bit clumsy, and your correct description seems overly long. 'Small Circle' might work, but could be confusing.

I'd better stop digging my hole any deeper and accept your wording as better. With Earth being oblate/pear-shaped I don't much fancy working out the exact surface distances from the pole.

Not giving up until I hear the 'answer'.

Taking question as stated; 12 dots, and no stipulation about folding the piece of paper cut out. With blackbelt origami I could make 12x11x10x9 = 11880 configurations. I'll not include size of square because that wold be insane and result in infinity.

I'm not inclined to illustrate all examples, but if anybody is mad enough to select and number any 4, I'll post an image for that case.

Not as many as that Kris:

The first can be any one of 12

The second can be any one of 11

..

..

Yes, but, it doesn't matter what order they're in, so it's (12x11x10x9)/(4x3x2x1) = 495

That did cross my mind, but say any 4 dots (call them ABCD) are used. Going clockwise, ABCD could be considered a different square to ADBC, etc. Nearest comparison is dice being either left or right handed. Division by 24 may be justified, but there is scope to argue it. The point may be moot if folding is accepted, because any given arrangement could be tweeked to give an infinite number of sizes for the square formed.

Even without folding creases, the suggested print out could be rolled into a cylinder allowing more squares than seem obvious. That doesn't yield a greater number, but is easier to visualize.

Perhaps the blog owners will relent with an answer one day. It's getting harder to suck any more marrow out of this. For reasons such as chirality, I'm sticking with 11880.