Freiman Sumset Puzzle (A ⊂ {0,…,9}, |A| = 4; given A+A, recover A)
Pick four distinct numbers so that their sumset matches the shown A+A.
150
Countdown
Target A+A (unique sums)
Your guess A (4 distinct elements)
Universe: {0…9}

How to play

  1. The system secretly chooses 4 distinct numbers from {0,…,9} to form A, and shows A+A (all sums a+b with a,b∈A, deduplicated).
  2. Click numbers below to build your guess A (4 numbers).
  3. Click Submit. If the sumset from your A matches the target A+A exactly, you win. Click New Puzzle to try another.
  4. You have 150 seconds. After time is up, you can keep experimenting but it counts as overtime.
A bite-size Freiman theorem

In additive combinatorics, Freiman's theorem describes the structure of sets with small doubling: if a finite set A has a sumset that is not much larger than A itself (e.g. over the integers |A+A| ≤ K|A| for a modest K), then A must be highly structured — it is contained in a generalized arithmetic progression (GAP) of bounded rank and size depending only on K. Intuitively: “if addition creates few new elements, the set is close to arithmetic structure.”

This game gives you A+A and asks you to reverse–engineer A — a playful echo of “structure from small doubling.”

Tip: Different sets A can yield the same A+A. Any matching A is accepted. To enumerate all solutions, click Check all solutions.