Floor Tiles: Newsletter Challenge (10/30/07)

For 81X63, there will be 103 tiles crossed. For 472X296, there will be 577 tiles crossed.

He must be skipping a few!

ROFL

C is the larger floor area 472 by 296, A is an area of 59 by 37 which is the Lowest Common Denominator of the large floor C, and where the line passes through a corner, B is assumed to be the position of the laid tiles 81 in the long direction by 63.

"If a straight line is drawn diagonally across the floor from one corner to the other, how many tiles will it cross? If the floor is 472 x 296 tiles"

NOTE: The position of B is not given, and the word floor is used in the singular.

The pattern of 3 singles and 5 doubles will repeat every 8 columns. 59/8 = 7 repeats, and the last three columns in area A will be 2 singles and 1 double.

A = ((3 + (5 * 2)) * 7) + (2 + (1 * 2)) = 95.

(81—59)/8 = 2 repeats and the last columns = 3 single and 3 doubles.

B = A + (((3 + (5 * 2))*2) + ((3 +( 3 * 2)) = 95+35 = 130.

C = A * 8 = 760.

Regards JD.

From my experience floor tile is normally square. The only time I can see them being any other shape is specialty cut.

A rectangular floor is laid out made up of square tiles.

The shape of the tiles has been specified in the first line of the problem.

Yes, I'm responding to the picture with all the tiles being rectangular and the line cutting through each one equally.

You are correct, it did specify square tiles, however, I don't really think it matters whether the tiles are square or rectangular. As long as they are all oriented the same way, as he did in his graph, the effect is the same. Try it yourself.

Agreed: rectangular tiles will behave the same - provided that the grouting (if any) is proportional to the tile in the same direction (11/10 for pedantry?).

[I recognise that the comment and the diagram are not strictly related - I think the picture showed sections where the line went through corner tiles]

The drawing is not of tiles but floor areas in which the tiles are laid. Sorry if it caused you some confusion.

Regards JD.

In my boyhood home we had a bathroom that had a tiled floor. Each tile was either square or a rectangle one square wide by two squares long. The rectangles were clearly not specialty cut as they had bevelled edges that were round in such a way as to indicate being formed in a mold. The squares and rectangles were also complementary colors in contrasting shades.

These squares and rectangles were then laid in a variety of fanciful patterns patterns that interlocked with adjoining patterns for an almost infinite variety. I always enjoyed my otherwise wasted (no pun intended) sitdown time observing these patterns. Of course back then we did not have iPods, Gameboys, tiny TV's, etc. so what else was a kid to do, read a book or a newspaper? <grin>

Maybe it's the same floor, but the replacement tiles are smaller and rectangular? (Fortunately, this doesn't affect the answer - assuming all rectangles to be in the same orientation)

when you make a grid 7 x 9 you pass through 15 blocks. Your diagonal line will go through 9 the long way and 7 the short way. Don't count the first square twice that's why you subtract 1 from the equation 7 + 9 - 1 = 15. Now when we double the lenghts to make the next grid we go through 16 blocks. So a grid that is 14 X 18 you will go through 15 + 16 = 31 blocks. So a grid that 63 By 81 is 63 + 81 -1 = 143. If the floor is 472 x 296 you will cross 472 + 296 - 1 = 767. This works with any grid that is not a square.

9 X 7 grid, Diagonal passes through 15 Tiles as mentioned. (you get a pattern of 2,2,2,3,2,2,2 repeated every 9 tiles across the horizontal axis, and 7 tiles on the y axis) 81/9 =9, 9 x 15 =135 tiles.

In the Larger Grid 472 x 296, the pattern is 2,3,2,3,2 repeated every 8 tiles across the x axis and 5 tiles on the Y. This gives 12 tiles for every 8 across. 472/8 =59,

59 x 12 =708 tiles in larger room.

CK.

Scratch the last part, I double checked using autocad, and concur with previous posts of 760 tiles crossed in larger room, 95 tiles every 59 squares across, 472/59 =8,

8 x 95 = 760. (I am using the figure of 95 tiles crossed by diagonal, you could argue that this figure should be 97 because of the intersection point of 3 tiles, this gives a total of 97 x 8 = 776 - 2 = 774 in total crossed.

ck

The line crosses 81 tiles; in the larger floor, the line crosses 472 tiles.

If you get a math linear equation

y=7/9x, x=0 to 81

the max number of tiles to be crossed is 81 (all of them in x axis)

When you get y=integers you are not crossig any tile. You are in the intersection border.

This occours 9 times (x=9, 18, 27, 36, 45, 54, 63, 72 and 81)

Then you cross 81 - 9 = 72 tiles

the max number of tiles to be crossed is 81 (all of them in x axis)

Not so, see the graph in ddk's post, #12. You will often see 2 tiles in the same column, which means the both tiles have been crossed without incrementing the x-value.

That´s right.

the increment with the same integer happens 9 times; so the right solution is:

81 - 9 - 9 = 63 tiles, with 9 intersections and 9 increments with same integer.

This solution seems awfully one-dimensional...

(sorry; couldn't help myself! )

The other direction would mean lay 81 tiles in the direction to your right and then lay 63 tiles the other direction which would be to your left, so you have a line of tiles 144 long. Then draw your line diagonally across all the tiles from corner to corner. The answer could be 144 tiles. This is why we use Geometric Dimensioning the 63 tiles maybe should be perpendicular in reference to the 81 tiles.

just a thought

"just a thought"

Not a good one, though. I think our Guest is pulling our leg(s). Either he is a moron, or being deliberately obtuse to get a laugh. Obviously, the intent of:

"A rectangular floor is laid out made up of square tiles of the same size. 81 tiles are laid out in the long direction and 63 in the other."

is that there should be a matrix of 5103 tiles which is 81 tiles long and 63 tiles high (the "other" direction being at 90°, not 180°!)

How many floors have you seen tiled with just one long row of tiles? I guess it could be a really long hallway with a single square tile being the entire width of the hallway and the length of the hallway then would be 144 times its width. Yeah, right.

C'mon, who is it now? Fyz? Kris? ER? 3Doug? cr3? AH? Fess up!

Well, this will be certainly one to see what the official "Answer" is, since we have so much disagreement. Obviously, it is hard to argue with the draw and count method, if drawn and counted correctly!

But I think I understand the calculating method, and can put it into simpler, less mathematical terms.

If we can break up the grid into identical smaller grids with sides that do not have a common factor (and so cannot be broken into smaller identical grids), the diagonal of one small grid will cross a certain number of tiles, n. The larger diagonal will then cross z tiles where z = n x f , if f is the common factor between the small grid sides and the large grid sides. In other words the small diagonal is repeated f times.

In crossing any grid the diagonal line must cross into other blocks under three conditions. It will cross to the next tile vertically across a horizontal edge, horizontally across a vertical edge, or exactly through the corners. If you have the smallest possible identical grid, the small diagonal line will only touch a corner at the start and end. Therefore, to cross into the next tile it must cross either a horizontal or vertical edge, and every time it crosses an edge it will be in a new tile. If the grid is L tiles long and H tiles high, the line will cross L-1 plus H-1 edges, to still remain within the small grid. And so the line enters L+H-2 new tiles, and adding the first tile in which it started, the total tiles crossed in a small grid, n, is thus

n = L + H - 1 and the sum total for the whole large grid is z = (L + H - 1) x f

So, for the challenge questions of 63 x 81 with a common factor of 9, giving smaller grids of 7 x 9, the answer is z = (7 + 9 - 1) x 9 = 135.

and for the other question of 472 x 296 with a common factor of 8, giving smaller grids of 59 x 37, the answer is z = (59 + 37 - 1) x 8 = 760.

Hopefully, this will bring into the fold any non-believers who cannot fathom the higher mathematics involved in some of the other explanations.

I think that is the basis of a simpler (=> better) explanation than my first try (albeit on the same basis).

To rephrase the central section:

In each subsection, and starting on the diagonal at one corner:
When leaving a tile, the diagonal must cross either a horizontal or a vertical gridline. It must cross each internal gridline exactly once, which corresponds to leaving exactly
(L-1) + (W-1) tiles. It enters but does not leave the tile at the end corner, giving a total of L+W_1 tiles

one

I actually think thst the answer is as follows.

7 & 9 are prime numbers and as such cannot be reduced. Therefore the diagonal line must pass through one hroizontal and one vertical tile as it moves up each row. The only exception is at the end where it only passes through one tile. Therefore the number of tile = L + W -1 or 9+7-1 = 15 per each 9 x 7 block.

This work for any block where the numbers cannot be reduced ie a 5 x 3 block has 7 tiles passed through on the diagonal. A 2 x 2 block only 2 tiles on the diagonal as it can but reduced to a 1 x 1 (which as only 1 on the diagonal = 1 + 1 -1.

Hope this makes sense.

The solution is:

Nt=Nw*Int(Nl/Nw)+2*rest(Nl/Nw) where Nt, Nl and Nw are the Nr. of tiles crosed, Nr of tiles on long direction and Nw Nr of tiles on the with.

Ex1: Int(81/63)=1, rest(Nl/Nw)=Nl-Nw*Int(Nl/Nw)=18, Nt=63*1+2*18=99

Ex2: int(472/296)=1, rest(472/296)=176, Nt=472*1+2*176=824

No. I'm not betting.

To clarify: my results would always have been 135 and 760 as per the formula**.
95 (given in an reply to 3Doug, I think) was the number crossed in a 59x37 section of the 472x296 floor

**Numbers may be reduced by spacings between tiles (grouting?). If there are spaces, they may be further reduced by irregularity of structure, allowing additional effective corners to be created. They could also increase if the gaps at critical tiles are smaller than the average (so we lose internal corners)
I think(??) the formula was first presented by Math_Physics_Maniac. Most of my postings were intended to provide explanation for the formula for the internal sections.

BTW, you'd be hard pressed to make the spacings in the 472x296 floor regular enough to see this number of crossings in practice.

We can calculate the no. of tiles being crossed mathematically. Here is the formula.

Take out the maximum common factor of the two nos.(lengthwise and widthwise no. of tiles)

So, 81 & 63 ----max. common factor is 9

i.e., 81 = 9*9 & 63 = 9*7

Now add the remaining nos. and substract 1 from the answer.

i.e., 9+7-1 =15

Now multiply this answer with the max common factor and you get the answer to your puzzle.

i.e., 15*9= 135 tiles.

The line drawn digonally accross the conners of the floor will cross 135 tiles.

Through same calculation, the answer for 472*296 is

8*59 8*37 59+37-1=95 95*8=760 tiles.

Floor tiles 472 x 296

Divide both 472 and 296 by 8, yielding 8 sets of 59 x 47 tiles.

The two numbers 59 and 47 cannot be both divided by another whole number.

The number of tiles crossed = 472 + 296 -8 = 760.

In the 59 x 47 grouping, only one tile multiplied by 47/59 equals

a whole number, namely 59 x 47/59 = 47; similary whole numbers are generated

when using the ending tile numbers in the remaining 7 groups, that is 118,

177, 236, 295, 354, 403, and 472.

This rule also applies to the 81 X 63 arrangement. Divide each by 9, yielding

9 sets of 9 X 7. The number of tiles crossed is 81 + 63 -8 = 235.

Obviously, the number of tiles crossed will be equal to the tiles fitted into length of rectangle.i.e 81 tiles.if the floor would have been square then the answer would be equal to no. of tiles on breath (other)side.

If floor is 472 x 296 tiles, then answer will be 472 tiles.

I think my answer should be right logically.

Thanks

"I think my answer should be right logically."

Then think again, Vaishkau, because you are neither right nor logical. If you think this is logical, then you must not know the definition of "diagonal" or some of the other terms used in the question. Only a line parallel to the base line which has 472 tiles will cross only 472 tiles. The diagonal is on an angle from one corner to its extreme opposite corner, and thus crosses many more tiles than a parallel line would.

Even those who suggested using Pythagoras were closer to the right answer than you are!

Sorry in advance if this seems too "confrontational", I am just getting tired of these STUPID answers, when the correct answer has been given so many times already! Yes, I made a mistake, but early on when there were not a lot of other answers to check mine against. When my error was pointed out I accepted that I erred and moved on. I hope you do also.

re. getting more confrontational - do you think it might be time for a holiday?

re. getting more confrontational - do you think it might be time for a holiday?

We just had a holiday, last night, Halloween. Talk about confrontational! At my door I confronted all sorts of creatures demanding sugary treats; Ghosts, Goblins, Witches, Freaks, Disney Princesses, The Simpsons, Zombies, and the scariest monsters of all, sugar-crazed Teenagers!

Oh.... British "holiday" a.k.a. (in American English) as "vacation" ...

...yes, I do!

No we don't. My esteemed chum STL- let me try to enlighten you on Brit habits. We don't do the 'trick or treat' thing although we have inherited the habit. Increasingly it occurs, but not with much approval. Mainly we celebrate 'Bonfire Night' ( aka Guy Fawkes night/November 5th). This involves setting off fire-works, meeting close family for food etc. It does tie in to a lot of historical stuff about the pope and that, but generally it's about having a laugh and setting off explosive stuff. I once burnt my hands real bad ( though that is a long story), and a certain amount of care is needed. I know of at least one other member who has a certain expertise in such things. My own experience is limited to such stuff as dynamite and certain fertilizer based stuff, but well, accidents happen. Wah !

Fyz and Kris,

Yes, I am well aware that Brits do not traditionally celebrate "All Hallows Eve". However, in the USA it is a "big deal" for the kids to dress up in costume, often attending several costume parties in the week or so before Halloween. Public Schools now try to downplay it somewhat because of the implied connection to religion and vandalism/violence (funny how those got tied together!), but it really isn't about either, although some use it as an excuse. It is a way for kids to come to terms with scary stories and things that go bump in the night, finding out that its mostly just put on for show, and to enjoy some sweet treats. How the young ones DON'T make the leap from there to "There is no REAL Santa/St. Nick/Father Christmas" I don't know, I guess its self-interest (toys and other gifts, treats, etc.)

Some people (especially grownups who should know better) try to celebrate Halloween for the occult implications. As far as I am concerned they can just go to h......well, you know where! The typical suburban neighborhood will usually have dozens or more small groups of 2-3 or even 10-12 kids banding together with one or more adult/parents accompanying them, especially if the kids are smaller, going door-to-door. This year my wife walked around our neighborhood with our 3 children while I stayed at home to give out candy. The usual call by the approaching kids (if you are sitting outside or when you answer the door) is "Trick or Treat" (as once upon a time Halloween was more of an excuse for mischievous tricksters to prank their neighbors, but first allowed them the chance to buy their way out with a bribe of sweet treats). These days the only Trick that usually happens is for one of the kids to tell a joke, sing a song, or otherwise practice a stunt, usually related to the costume they are wearing. But, whether they present a "trick" or not, they usually get their "treat". And the treats can vary a lot also, depending on the relative wealth and/or generosity or just plain quirkiness of the host. On of our neighbours hands out 6-packs of soda cans, while most just give out bite-size versions of popular candy bars. Popcorn balls made with caramel are popular also, as are caramel covered apples rolled in nuts (on a stick). Jawbreaker candies, lollipops, nuts and salty snacks, and even small toys are often found as well. Some kids also collect money for UNICEF at this time, although that is not as popular as it once was.

BTW, fireworks usually do not play much of a role in our Halloween. I heard not one POP last night. We usually save our pyrotechnics for Independence Day (July 4) and New Year's Day.

Yes, All Hallows, or All Saints Day, is still part of the Catholic Church calendar, but is not such a big celebration as Christmas, Easter, of even Thanksgiving, a family holiday with religious overtones (giving thanks to God). Many countries celebrate a Thanksgiving, or Harvest festival, but in the USA our tradition hearkens back to some of the first colonists, the Pilgrims, a Puritan sect from England, who settled near Plymouth, Massachusetts, founding one of the first permanent settlements in the English colonies in America which became the United States. During the time leading up to our Thanksgiving, kids will produce Pilgrim-themed artwork, sometimes even dress up as Pilgrims, or put on plays about the Pilgrim's First Thanksgiving. It has special significance for us because it celebrates the friendship of the Pilgrims with their native American neighbors (despite subsequent sour relations between various native and immigrant groups) and gives hope that all potential enemies can come together in friendship and fellowship and sharing in the bounty of the land.

As I mentioned, Public Schools now try to downplay any connection to these so-called religious holidays, but people and traditions being what they are, the names get changed to be politically correct and the celebrations go on. So now, instead of Halloween or Thanksgiving celebrations, we have "Harvest parties" or "Fall celebrations" and instead of Christmas, or Hanukkah, there is a "Winter party" or "Holiday Season Celebrations".

So let me be the first to wish all of you Seasons Greetings on this November 1. Remember, only 53 more shopping days until Christmas!

***** heck that was long man ! I wish that I could match you in volume, but I fear I cannot . If I was not preoccupied with my 'bath' elsewhere I would try to engage you in this. Have fun trying to educate the good people here.

I guess I should get my own blog! But then, who would be my captive audience? <grin>

That is a fantahbous post STL. Just to make it clear, I never indulge in 'anon' posts in the Challenge Quesions. I enjoy posts made by you and Fyz,-The level of detail and atention to subject is bar non. When I take the *** I make Iit very clear that I am 'moi' taking the ****. I've never hurled isults to hurt you guys.

Why do engineers get confused between Halloween and Xmas?

.

.

.

Because, Oct31 = Dec25

Sorry: this joke has limited mileage in normal company.

They probably get confused because the stuff made on 14th March had gone a bit crusty. It's only programming types who go mad at this time of year.

I get it - what a coincidence!

I won't tell yet in case someone else is trying.

Maybe you should have said software engineer. (obvious hint)

Note that this is a mathematical problem not a technical one. Perhaps there's a recurring pattern with differently sized rectangles.....dependent upon both the long and the short side. Increasing by increments of one each the number intersected, or so it was found was usually one more than the longest side or the longest side itself. In the case of 9x7 that's not true. However there is probably a different relationship as the long and short sides stray from each other and differences in patterns, if any exist, are magnified.

Can it be 143?

I was just guessing and the hypothesis for a "m x n" matrix I could come up with is as below.

No.of crossing = (m + n) - 1, if (m < n) or (m > n)

No,of crossing = m, if (m = n)

Any comments!!

No. That only works when m and n are co-prime.

Try dividing into smaller rectangles (9x7), work within these rectangle, and see where that leads

If m=n, then the minimum unit is 1 x 1, 1 + 1 = 2 tiles crossed = (2 - 1) x m.

It has to work for all cases.

rich

Once again I must say thanks to ddk. His graph caused me to go back and take another look at my count. Upon the proverbial closer inspection, I discovered a counting error on my account. My corrected count stands at 95 for the sample grid on the larger floor, giving 760 for the number of tiles the diagonal crosses.

If I hadn't discovered this, I probably would have flipped a coin....;-)

Too early to Jack it in, I think. A little work on the relationship between the diagonal leaving squares and crossing grid-lines (au STL) would clarify all. Apart from being more certain (as you and many others have shown), it's also quicker.

Well done Folks, I think we are all in agreement that we've cracked this one.

I hope you are right, but I will bet that postings with WRONG answers will continue to trickle in, especially from anonymous Guests and others who don't bother to read past the original question before posting!

Anyone for a pint a of Guinness?

That reminds me of the story of the Presidents of 5 major breweries who had gathered at the bar during an International Brewing convention. The President of Anheuser-Busch (St. Louis) stepped up to the bar and said, "Barman, give me a pint of Budweiser, the 'King of Beers!' "

Not to be outdone, the President of Coors drawled, "Give me a Coors, y'all, 'brewed with pure Colorado Rocky Mountain spring water'!"

The president of Fosters bellowed, "Give me a Fosters, mate. That's 'Australian for BEER' !"

The president of Becks said very simply and very Teutonically (sounding a bit like Arnold Schwarzenegger), "Give me a Becks, ja?, because 'Life is what you CHOOSE'!"

The president of Guinness, in a light Irish lilt, quietly said, "I'll have a Coca-cola, please."

"What?!", roared the president of Foster's, "You are not even going to order one of your own beers?"

"Oh, 'tis that what you were doin' now? I tought you was all orderin' soft drinks!"

Hi MPM,

This problem had an unexpected solution. Now, when someone is trying to figure it out, I can come out with the answer withou so much as a pencil (or a calculator for that matter). Yeah, like that'll ever happen.

Thanks to all of you who (intelligibly) contributed!

By the way, I was noticing the other photos. Is lead-free solder that much of a problem? I thought that the problems were mainly in sourcing the lead-free components.

-m-

This probably deserves its own thread, but, I think that lead free solders typically have a higher melting point than traditional 50/50 or 60/40 tin-lead alloys. They also may be mechanically weaker, and growth of "tin whiskers" leading to circuit shorts has also been a problem.

Of course you could use lower melting point "Wood's Metal" to solder, but it might not hold up if your circuit overheats at all, or even on really hot days inside an electronics cabinet!

ROFL!

PS-WikiP says that the lead and cadmium in Wood's Metal make it toxic, and a non-toxic choice might be "Field's Metal", which has an even lower melting point!

Hi Mikey,

The photos which I have posted there show a phenomenon called hot tearing when a SAC305 lead-free through-hole joint cools down on an electronic PCB. It is apparently an acceptable cosmetic defect that does not affect long-term joint reliability. It does look quite horrible when viewed at magnification.

There is a separate thread for this discussion at:-

http://cr4.globalspec.com/thread/12064#

If 81x63 tiles are laid out,the number of crossed tiles is:

((27+21)-1) X 3 = 141

And if there are 472x296 tiles the answer is:

((59+37)-1) X 8 = 760

Check it!

I was wondering if there was any simularities in node pattern between the different numbers and came up with these! They have not helped me fathem out an easier way tho! Thought you might like to see my results! These are the small grid layout, if you want our question, look at the last grid and remove the top two rows!

The answer is of two separate floors and two separate lines? when the puzzle only stated one floor and one line crossing two separate areas.

Regards JD,

Well no good winging, I suppose it's going to cost me a guinness, see you all down at the pub.

Regards JD.

How do you think I feel? I was betting on 0.

bob

Guiness or tiles?

Regards JD.

C, All of the above.