Secret Santa · 10,000,000 simulated games

Secret Santa with 3 couples: what is the chance the hat draw gets stuck when partners can't draw each other?

6 people draw names from a hat in turn. Anyone who pulls their own name or their partner's puts it back and draws again. How often does someone reach the hat and find only names they can't take?

36.111%exactly 13 in 36
simulated: 36.0998% ± 0.0298% (95%), 3,609,981 of 10,000,000 games

In a Secret Santa hat draw with 3 couples, where you redraw your own or your partner's name, the chance the draw gets stuck is 36.111%. Worked out exactly and confirmed by 10,000,000 simulated games that you can replay on this page.

What the computer did

Games: 10,000,000 (1,000 chunks of 10,000, in 1 segment)
Compute: 0.01 CPU-hours; 0.00 hours on arm64, 16 cores, Darwin 25.5.0
Engines: Python 3.14.8; every segment is spot-checked by replaying chunks in the other language
Ran: 2026-10-31T14:54:21Z to 2026-10-31T14:54:25Z
Seed: 3456958452 (from this page's address); chunk k uses xoshiro128** seeded by splitmix32(seed ⊕ k·0x9E3779B1)
Exact method: every possible sequence of draws, weighted by its probability (exact recursion)
Code: sims/rng.py sha256 b81126f5f114cdf1…
sims/kinds/hatdraw.py sha256 4de49c14ddd0d8a9…
site_src/sims.js sha256 5cd91ae0e3f8246e…

All 1 segments of this run

#	Chunks	Chunk size	Games	Engine	Finished	Code (sha256)
0	0–999	10,000	10,000,000	Python	2026-10-31	b81126f5 4de49c14 5cd91ae0

Three checks your browser runs

The totals add up. Adding up every published chunk…

It agrees with the exact answer. Your browser is working out the exact answer by a separate method…

The chunks are real. Pick chunks below and your browser replays them from the published seed. Each must match to the last game.

Replay any chunk

The run is split into 1,000 chunks of 10,000 games. Each chunk has its own random stream, built from the seed and the chunk number, so any one of them can be replayed alone. Your browser picks which ones at random, so the run could not have been prepared for your choice.

replayed by youreplayed by other readersnot replayed yet

or chunk

No chunks replayed yet. Each cell above is a chunk; replayed ones turn solid.

What your replays rule out

Suppose someone had made up a share of the chunks instead of running them. These are the chances your replays so far would have landed on at least one made-up chunk.

–if half the chunks were made up

–if 1 in 10 were

–if 1 in 100 were

Breakdown against the exact answer

Outcome	Simulated	Exact	Off by (std. errors)
The draw gets stuck	36.0998%	36.1111%	-0.74
… at the last person	33.3184%	33.3333%	-1.00
… before the last person	2.7814%	2.7778%	+0.70

Independent simulations by local models

Each model below was given only the rules of this family of questions, in plain English, and asked to write its own simulation. It never saw our code. Its program was run in a sandbox on two test cases and compared with the exact answers.

gemma4:26b agrees z = +1.00; z = +0.23
gpt-oss:20b agrees z = +0.87; z = -0.75
qwen2.5-coder:7b error error; error
qwen3.6:27b-mlx agrees z = +0.87; z = -0.75

See the code each model wrote

What a match proves, and what it doesn't

A matching chunk proves those games come from the published code and seed. Nobody could guess every count exactly without playing the games.
It does not prove the code plays the game described. That is what the exact answer and the independent simulations are for.
The run (Python) and the replay (JavaScript) were written by the same author, so they are two copies of one program. Their agreement shows the numbers were computed, not invented.
One reader replaying a few chunks rules out large fakes, not small ones. Every reader's replays are added up, so together they cover the ledger.