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Playing with Chips - Problem-A-Day

joshua_greenAug. 5th, 2009 09:20 pm Playing with Chips

    Suppose you have a stack of N poker chips.  If N ≥ 2, you split the stack into two smaller stacks of sizes m and n with both new sizes positive integers and, of course, m + n = N.  While you're at it, note the product mn.  You can now repeat the process, splitting any stack(s) of size ≥ 2 into two smaller stacks and noting the product of the two new sizes.  After N - 1 steps, you will have N stacks of size 1 each, hence no more splitting will be possible, and you will have noted a bunch of products.  Add all these products together.  As an example, if N = 5, you could follow the final steps:

  1. Split the stack of size 5 into stacks of size 2 and 3 and note the product 2 × 3 = 6.
  2. Split the stack of size 2 into stacks of size 1 and 1 and note the product 1 × 1 = 1.
  3. Split the stack of size 3 into stacks of size 2 and 1 and note the product 2 × 1 = 2.
  4. Split the stack of size 2 into stacks of size 1 and 1 and note the product 1 × 1 = 1.
  5. Add all the products together, arriving at 6 + 1 + 2 + 1 = 10.
Of course, there are choices along the way, as there may be more than one way to split a stack into two smaller stacks.  Show, however, that the final sum of the products (10 in the example above) is independent of the choices made (though it may depend on N).

Current Mood: amusedamused

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Comments:

From:rdore
Date:August 6th, 2009 04:55 am (UTC)
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This counts the number of ways to pick two elements of {1,...,N}. Pick one from each piece, or pick both from one half or the other.
From:joshua_green
Date:September 20th, 2009 04:39 pm (UTC)
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Of course!