Suppose you have 27 weights weighing: 1², 2², 3², 4², ........, 25², 26² and 27² grams respectively.

(a)How can you group them into three groups so that each group has the same weight ?

(b) Is it possible to divide it into more than three groups satisfying the same conditions ?

Ok... I was working around with squares, and I think I know why that would work.

When you take perfect squares, you can split them up into consecutive odds.

Any x³ weights can be split into piles of x using this method. Take 8 weights into 2 groups. (Put them into a 'cube' shape, where the left side is one group and the right side is another group)


Top level
01 04
16 09

Bottom level
36 25
49 64

So if you put in 1 through 16 in using the method shown (AB BA), it comes out

1, 1+3,
1+3+5+7, 1+3+5,

it can be seen that the left row has an extra 7 and the right row has an extra 3. When you slide the next group (the 25 through 64) over one, you switch these two so they come out even.

Taking out 1+3+5+7 from each number in the bottom layer gives:

9+11, 9
9+11+13, 9+11+13+15

Now the right row has an extra 15, and the left row has an extra 11. So together, the left side on the top level is 4 more, and the right side on the bottom level is 4 more and the rows 'cancel' each other out. This means that the difference between the two is 0, and so they are equal.

This idea applies to a 3 by 3 by 3 grid too. Rotating the highest in each level and each row works as well.

You can conclude that (on the top level of the 3 by 3 by 3), the left side has a 9+15+17, the middle has a 3+9+11, and the right has a 3+5+15

The next floor these values will shift one time to the right, and then they shift one time to the right again. This means that all the rows will have the same parts, and will be equal.

(I hope I explained it as best I could.)

Posted by Gamer on 2003-05-31 10:09:39