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Friday Puzzler -- Chocolate Bar

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Friday Puzzler -- Chocolate Bar

#1

Friday Puzzler -- Chocolate Bar

Alex Y

A chocolate bar is scored to allow it to be broken into smaller rectangles. The scoring is such that the "grid" of chocolate rectangles is m pieces in one direction and n pieces in the other.

If you break the bar using only straight-line breaks in the original bar and in the resulting pieces until the bar is broken into all the individual rectangles, how many breaks must you make? Each break is of a single piece of chocolate--no stacking the pieces and breaking multiple pieces at once.

Extra credit for the "aha" solution to this one -- I got it only by brute force, but maybe even if you do that, you can look for the "aha".

Re: Friday Puzzler -- Chocolate Bar

#2

Re: Friday Puzzler -- Chocolate Bar

Ed in Leaside

I think the aha is that it doesn't matter whether you break m or n first ...

Aside: speaking of chocolate, m, and n. I once knew a typesetter who's company was called Em & Ens. :-)

Re: Friday Puzzler -- Chocolate Bar

#3

Re: Friday Puzzler -- Chocolate Bar

Alex Y

I think the aha is that it doesn't matter whether you break m or n first ...
And in fact, you can toss a coin to decide in which direction to make each break (as long as both directions are available). But why is the answer what it is?

Aside: speaking of chocolate, m, and n. I once knew a typesetter who's company was called Em & Ens. :-)
LOL!

Re: Friday Puzzler -- Chocolate Bar

#4

Re: Friday Puzzler -- Chocolate Bar

Ed in Leaside

You start with one piece, and each time you snap a score you have one more piece.

Re: Friday Puzzler -- Chocolate Bar

#5

Re: Friday Puzzler -- Chocolate Bar

Larry Barrett

If you first break in the n direction you break n-1 times and end up with n strips. Then, break each of the n strips in the m direction m-1 times. Total is (n-1) + n(m-1) = n-1 + nm -n = nm-1.

If you first start in the m direction the result is (m-1) + m(n-1) = m-1 + mn -m = mn -1. Same as before.

Re: Friday Puzzler -- Chocolate Bar

#6

That's it! 


Re: Friday Puzzler -- Chocolate Bar

#7

Re: Friday Puzzler -- Chocolate Bar

Alex Y

That's exactly the math I followed. Ed's answer is the "aha" I never saw -- no algebra required, and it covers any sequence of breaks.

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