MathLabs

Problem 3

There are 4n4n pebbles of weights 1,2,3,…,4n1, 2, 3, \ldots, 4n. Each pebble is colored in one of nn 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 5 of 6: Read off the two piles
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In plain words

Since a string is a single edge, its colour sends both of its pebbles to the same pile, and the vertex balance from the previous step controls how many pebbles of each colour land in each pile.

Pileblue={pebbles on a blue string},Pilegreen={pebbles on a green string}.\text{Pile}_{\text{blue}}=\{\text{pebbles on a blue string}\},\qquad \text{Pile}_{\text{green}}=\{\text{pebbles on a green string}\}.
Detailed analysis

Put every pebble whose string is coloured blue into one pile and every pebble whose string is coloured green into the other; because each string is a single edge, its two pebbles always land in the same pile. By the previous step, each box (colour) has exactly two pebbles on blue strings and two pebbles on green strings, so each pile contains exactly two pebbles of every colour.