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Stamps of 3 and 5

Issue 8 Core Published 26 Aug 2026 By The Chalkboard Desk

Tier Core — Stamps of 3 and 5

The problem

A post office sells only -cent and -cent stamps, in unlimited supply. What is the largest amount of postage that **cannot** be made exactly?
  • You may use any non-negative number of each stamp.
  • The amount must be made exactly, with no change given.

Take a minute

Reveal hint 1
Work modulo : once you can make three consecutive amounts, you can make every larger amount by adding -cent stamps.
Reveal hint 2
Which consecutive triple of amounts can you make?

Solution

Answer cents.

Derivation

  1. List the small amounts.

  2. Find three consecutive makeable amounts.

  3. Extend by induction.

  4. Check what lies below.

  5. Confirm with the formula.

  6. Count the gaps.