Posted on Mar 16, 2026
Monthly Mindbenders (March 2026). Based on a problem from the 1972 International Mathematical Olympiad.
You challenge your bestie to the following game. She chooses 10 distinct integers between 1 and 100; you try to find two disjoint, nonempty subsets of her 10 numbers that have the same sum.
With best play, who will win?
You will win. No matter which 10 distinct integers she picks, two disjoint nonempty subsets with equal sum are guaranteed to exist.
There are 10 numbers, so there are \(2^{10}=1024\) subsets in total. Every subset has a sum between 0 and the sum of the 10 largest possible choices:
\[ 100+99+\cdots+91=955. \]
So the possible subset sums are in \(\{0,1,2,\dots,955\}\), which gives only 956 values. Since 1024 subsets exceed 956 sums, two different subsets must share the same sum.
If two different subsets have the same sum, their symmetric difference splits into two disjoint nonempty subsets with equal sum. Therefore you can always win.
Click to generate 10 distinct integers or enter your own. The generator will find disjoint subsets with equal sum.
Numbers: —
Subset A: —
Subset B: —
Common Sum: —
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