The Thorp Shuffle
In 1973 Edward Thorp, the mathematician who beat blackjack, described a shuffle you can do by hand. How many times you must repeat it before the deck is truly random was called the longest-standing open problem in card shuffling.
1 · Shuffle it yourself
Cut eight cards into two halves. Line up the halves, so each card has a partner. For every pair, flip a coin: heads, the left card drops first; tails, the right card does. The pairs, in order, make the new deck.
Each shuffle uses four coin flips, so it can produce only sixteen different results. That limit is the whole problem: eight cards can be arranged in 40,320 ways, and a few shuffles cannot reach most of them.
2 · One card versus the whole deck
This page now follows every possible outcome at once: all 40,320 orders of eight cards, and exactly how likely each is after each shuffle. Slide through the shuffles and compare two questions. Is any one card random yet? Is the deck?
Exact, not simulated: the page tracks the probability of every order after every shuffle. “From random” is total variation distance, the measure the paper uses. One card is the worst of the eight cards on its own. After three shuffles every card is perfectly random by itself, while most orders of the deck are still impossible.
3 · How the bounds grew
For a deck of 2d cards, each result below says how many shuffles are enough. Choose how many times the deck can be halved, and compare.
| Year | Who | Constant | For this d | On a log scale |
|---|
Orders of growth only. The 2005–2009 bounds hold up to constant factors their papers do not state, so the numbers here compare shapes, not real shuffle counts. The 2026 bound is the first that grows only in proportion to d. With its stated constant of 1,600 it falls below d3 only once d passes 40 — a deck of more than a trillion cards. The lower bound says no method can do better than about 2d.
4 · What is claimed, and what is checked
| Statement | Status |
|---|---|
| The number of shuffles needed grows in proportion to d (main paper, 14 pages) | Checked by machine, in Lean |
| 1,600d shuffles are enough, as decks grow without limit | Checked; “not a proposed sharp constant” |
| About 2d shuffles are necessary | Simple counting argument |
| 24d shuffles are enough (anonymous draft, October 10) | Not reviewed; “local draft, not yet published” |
| The proof as a whole, across twelve manuscripts and 683 pages | No specialist review found |
Fifty-three years
- 1962Thorp’s Beat the Dealer shows card counting can overcome the house edge in blackjack.
- 1973Thorp describes his shuffle while studying nonrandom shuffling in the game of Faro.
- 1992Bayer and Diaconis answer the question precisely for the ordinary riffle shuffle.
- 2005Ben Morris gives the first real bound, quoting Diaconis: the “longest-standing open card shuffling problem.”
- 2009Morris reaches d3. With Rogaway and Stegers, he builds encryption for card and social-security numbers on the shuffle.
- Oct 6, 2026OpenAI releases a machine-checked proof that the right growth is proportional to d.
- Oct 10, 2026An anonymous draft re-tunes the constant from 1,600 to 24. No other public response found.