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How Often Does a Perfect Shuffle Repeat?

Issue 12 Core Published 6 Sep 2026 By The Chalkboard Desk

Tier Core — How Often Does a Perfect Shuffle Repeat?

The problem

A deck of cards is cut into two piles of and perfectly interleaved — the top card of the deck stays on top (an 'out-shuffle'). If you repeat this shuffle over and over, how many shuffles does it take before the deck returns to its original order?
  • The interleaving is perfect: the cards alternate exactly, without any of the randomness of a real shuffle.
  • The top card stays on top after every shuffle.

Take a minute

Reveal hint 1
Label the card positions to and work out where position goes in one out-shuffle.
Reveal hint 2
The answer is the multiplicative order of modulo .

Solution

Answer shuffles.

Derivation

  1. Find the position map.

  2. Justify the modulus.

  3. Ask for the period.

  4. Compute the order.

  5. Confirm no smaller exponent works.

  6. Compare with the in-shuffle.