- #1

- 22,097

- 3,282

## Main Question or Discussion Point

You have a pile of N stones. You do the following: you take a pile and separate it into two smaller piles, multiply the numbers of stones in these two piles, and write this number on the blackboard. You do that until there is N piles with only one stone in each. Then you take a sum of all numbers written on the board. What result can you get?

Maybe an example will be instructive:

You start with a pile of 100 stones.

STEP 1: We divide it in two piles with 70 and 30 stones, thus we write 70*30=2100 on the board.

STEP 2: We divide the pile with 70 into two piles with 2 and 68, thus we write 2*68=136 on the board.

STEP 3: We divide the pile with 2 rocks into two pile with 1 rock each, thus we write 1*1=1 on the board

STEP 4: We divide the pile with 30 rocks into two piles with 15 rocks each, thus we write 15*15=225 on the board.

STEP 5: ......

As you can see, it doesn't matter which pile you divide into two piles and it doesn't matter into what numbers you divide the pile. Still the eventual sum of all the numbers on the board will be equal!!

Maybe an example will be instructive:

You start with a pile of 100 stones.

STEP 1: We divide it in two piles with 70 and 30 stones, thus we write 70*30=2100 on the board.

STEP 2: We divide the pile with 70 into two piles with 2 and 68, thus we write 2*68=136 on the board.

STEP 3: We divide the pile with 2 rocks into two pile with 1 rock each, thus we write 1*1=1 on the board

STEP 4: We divide the pile with 30 rocks into two piles with 15 rocks each, thus we write 15*15=225 on the board.

STEP 5: ......

As you can see, it doesn't matter which pile you divide into two piles and it doesn't matter into what numbers you divide the pile. Still the eventual sum of all the numbers on the board will be equal!!