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An infinite two-dimensional pattern is indicated above.

The smallest closed figure made by the lines is called a unit triangle. Within every unit triangle, there is a mouse. At every vertex there is a "sweet". What is the average number of "sweets" per mouse?

Nimit
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    A vertex is shared by six triangles, and a triangle is shared by three vertices, so $1/2$? – Shuhao Cao May 11 '13 at 15:48
  • It is true but how exactly $1/2$? I mean to say how to "calculate"/ to prove? – Nimit May 11 '13 at 15:57
  • Because there are infinite number of triangles, no boundary, every triangle and vertex satisfy that relation. – Shuhao Cao May 11 '13 at 16:00
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    I have interpreted this way: one vertex has 6 mice around them so each mouse has $1/6$ vertex. Moreover, each mouse has 3 vertices around them, so $1/6 * 3$. i.e. $1/2$. Thank you, @Shuhao Cao. – Nimit May 11 '13 at 16:13
  • Ha, no problem. :) – Shuhao Cao May 11 '13 at 16:16
  • You need to specify what "average" means, and there are several ways of doing it that make sense. One is to define finite boards and find the limit as the board size goes to infinity. Another is to fix a $k$ and require that each mouse must eat a sweet that is within distance $k$ of it (where the distance must also be defined). Both give the same answer for any sufficiently large $k$ and reasonable definition of distance. – user21820 Dec 29 '13 at 09:10

1 Answers1

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The answer is obviously 1.

Since the pattern is infinite, there's a countably infinite number of mice and a countably infinite number of sweets. Thus there's a bijection between the sets of mice and the sets of sweets, hence there's exactly one sweet per mouse and one mouse per sweet.

Wait a moment... I made a mistake. I forgot to take into account the insatiable hunger these mice have for sweets - each single mice eats all the sweets which lie on a single line. Luckily, since there are countably infinitly many parallel lines, each mice gets its own full line to eat, thus there are actually are countably infinitely many sweets per mice.

I do hope that after eating all those sweets, one of the mice will have the decency to eat this exercise too...

fgp
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  • Nah, it's half... each vertex is shared by 6 triangles... with 6 mice. Each mouse is only surrounded by 3 vertices. 3/6 = 1/2. 0.5 sweets per mouse. – Albert Renshaw May 16 '13 at 06:28
  • I am curious though if there could be a chain reaction... for example 1 mouse gets stingy and steals a little more than half of a sweet... the opposing mouse is now feeling left out so he goes and steals some sweets from a second vertex so that he too also has an extra sweet... This wouldn't work in a closed system but in an infinite system could they keep stealing a little more from the next one? (I haven't checked this at all, just a thought, I'm not going to go back and check either because I'm hitting the hey! Good night!) – Albert Renshaw May 16 '13 at 06:32