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 ?

(In reply to re(2): First answer: by Charlie)

My idea is because you are going forward (A to B to C) within the set, and backward (C to B to A) for the set's beginning number. (So it's ABC CAB BCA) (see the bottom for more explanation)

Also, the numbers, when put in a cube shape (the row, then depth... the tenth letter is below the first) forms a "magic cube" (like a 3 dimentional magic square)

ABC
BCA
CAB

CAB
ABC
BCA

BCA
CAB
ABC

The way I came up with this is by balancing the largest and smallest number. I started by doing the ABC CBA ABC CBA... thing. Then I realized that was a "period" of 6, and 6 wasn't divisible by 27. So I did 9 instead, but noticed that the largest one of the 9 always went into C. So I shifted the sets of 9 around the opposite way, to stagnate the largest number.

Other than my extremely loose explanation, I would be interested in a proof as I don't know why it would work.

Posted by Gamer on 2003-05-31 09:24:15