Date: Mon, 25 Jan 1999 15:13:44 -0700 (MST) Sender: Roger How many cannonballs can be stacked in a triangular pyramid?
Hi Roger - 1 if the pyramid has 1 layer
- 1+3 if the pyramid has 2 layers
- 1+3+6 if the pyramid has 3 layers
- ...
- 1+3+6+...+n(n+1)/2 if the pyramid has n layers
There is a nice way to exprress this number which you can see as follows. Suppose that n+2 people are standing in a row and you are to walk down the row and select three of them. You can do this in ways.
The third person selected could be person number 3, person number 4, person nunber 5 ... up to person number (n+2). - If the third person select is number 3 then the first two selected were 1 and 2 which could only be done in 1 = T(1) way.
- If the third person select is number 4 then the first two were selected from 1, 2, and 3 which could be done in = T(2) ways.
- If the third person select is number 5 then the first two were selected from 1, 2, 3 and 4 which could be done in = T(3) ways.
- ...
- If the third person select is number n+2 then the first two were selected from 1, 2, 3 ... n+1 which could be done in = T(n) ways.
Cheers
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