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Posts Tagged ‘puzzle’

Caution: Open your (junk) mail before shredding. Otherwise, you may find yourself trying to put together a jigsaw puzzle, like this one:

Shredded Bills

Oops.

An envelope with a survey came in the mail a couple of weeks ago, and while I usually open those sorts of things, Al put the unopened envelope in the shred pile before I did anything with it. Today while cleaning, Al was busy shredding when I heard him exclaim, “Oh my god, I shredded a dollar bill!”

He pulled out from the shred bucket the tatters of what used to be a form letter intertwined with strips of paper currency, and I found myself trying to determine if there was more than one bill in the mix. Putting the pieces back together was an exciting and challenging puzzle, but in the end since there turned out to be only two bills, it wasn’t actually that difficult. Al felt terrible about shredding the bills, but I found the whole incident rather amusing.

Now I’m not sure what to do with the two bills. I could attempt to tape the slivers together in their proper order, take the heavily bandaged bills to the bank and explain what happened, and ask if they would exchange the Frankenstein bills for wholly intact bills. Or I could mail the slivers to the Bureau of Engraving and Printing, which replaces mutilated currency, and request new bills. Or I could keep them as an art project. Decisions, decisions.

Oh, and then there’s the matter of the survey, which was the whole reason behind the mailing in the first place. I feel a little guilty about keeping the bills (or what remains of them) without returning the survey. The problem is, I don’t even know which company sent it because I didn’t bother to try to piece together the letter (I was having too much fun figuring out the dollar bills), and Al has already put the shredding in the recycle bin. I suppose I could sift through the bag, but it would be much harder trying to put the letter back together because of all of the other shredded white paper mixed in. So to whichever survey company sent me the mailing and entrusted me to participate, I express my sincere apologies. You wholeheartedly placed your trust in me and I let you down. I promise I’ll do better next time, in case you should decide to keep me on your mailing list.

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This week’s NPR Weekend Edition Sunday puzzle challenge went like this:

“Draw a regular hexagon, and connect every pair of vertices except one. The pair you don’t connect are not on opposite sides of the hexagon, but along a shorter diagonal. How many triangles of any size are in this figure?”

I suspect a mathematical formula exists to determine the answer. But as I don’t know of such a formula, nor do I know how to find one, I resorted to the manual method of counting.

This turned out to be not as simple as it sounds. I spent the better part of the week counting and re-counting and counting again, coming up with many different answers. Of course, there is only one answer, hence my determination to keep counting until I found it!

I began by drawing a hexagon and randomly connecting vertices, leaving just one pair unconnected (depicted here circled in red):

Then I began to look for triangles of any size. The first time I counted, I found 50. Figuring there could easily be some that I missed, I counted again more carefully and found 76. The third count yielded 84.

I then took a more careful and systematic approach to counting. It occurred to me that there is a symmetry in the figure that can be readily seen if the hexagon is rotated slightly:

Notice now that the left and right halves are mirror images. Knowing this made counting easier because if I found a triangle on one half of the image, that meant a corresponding triangle was on the opposite half of the image, making it less likely that I’d miss counting a triangle.

Starting with triangles comprised of once piece, I carefully labeled those triangles with a ‘1’. Next, I looked for triangles composed of two pieces and labeled those triangles with a ‘2’. I continued this method up to ‘10’, which is the greatest number of pieces I could find that formed a triangle. Adding up the subtotals for each number of pieces came to 80 triangles.

I went over all of the labeled images again and found some that I had mislabeled and some that I had missed. This time, adding the subtotals yielded 82 triangles. After double- and triple-checking my labels and arithmetic, I settled on 82 as my final answer:

[click image to enlarge]

On tomorrow’s Weekend Edition Sunday program, I’ll find out whether I counted correctly!

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