Magic & Math with Cards

Lecture 51 · Mathematics

Remainders, common multiples and matching cards

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What you will learn

Derive fixed-total matching from rotations and explain common multiples, coprime CRT and compatible noncoprime clocks.

Before this lecture

In this lecture

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  1. Read rotations as clocks

    • T≡r (mod m): same clock reading

  2. Recover the matching congruence

    • a+b≡−1 (mod m)

  3. Solve the common-offset case

    • T+1 multiple of lcm(4, 3, 2)=12

  4. Prove the coprime case

    • T≡2 mod 3; T≡1 mod 4 → T≡5 mod 12

  5. Find shared-divisor obstruction

    • Solution exists only if a≡b mod gcd(m, n)

  6. Prove generalized compatibility

    • Compatible ⇒ unique modulo lcm(m, n)
    • (m/g)k ≡ (b−a)/g (mod n/g)

  7. Apply the theorem within correct scope

    • Clock extension ≠ automatic unequal-packet trick

  8. Play and retrieve

    • Check shared information before searching

Companion interactive

Planned

Choose working move totals and solve or diagnose simultaneous clock constraints.

The interactive for this lecture is being built and will appear here.

Notes

Written notes for this lecture will appear here.

Sources

How these sources were used
  • Numberphile, “Chinese Remainder Theorem and Cards (Tadashi Tokieda)”: Rotation-matching inspiration; original checked opposite-order derivation and explicit distinction between lcm special case and generalized CRT.