Problem 3
There are pebbles of weights . Each pebble is colored in one of colors and there are four pebbles of each color. Show that we can arrange the pebbles into two piles so that the following two conditions are both satisfied:
- The total weights of both piles are the same.
- Each pile contains two pebbles of each color.
Step 1 of 6: Pair the weights symmetrically
In plain words
Pairing small weights with large ones by this rule always gives the same total, which is the key symmetry driving the whole construction.
Detailed analysis
For each k from 1 to 2n, tie a string between the pebble of weight k and the pebble of weight 4n+1-k; every strung pair has the same total weight 4n+1. As k runs over 1 through 2n, the partner weights 4n+1-k run over 4n down to 2n+1, so the strings use every one of the 4n pebbles exactly once, giving 2n strings in all.