The entry ticket to an exhibition can cost anywhere from 1p to 31p. You need…

Concept: To form every whole value from 1 to N using the fewest possible parts, where each part can either be included or excluded from a payment, split N following the binary (base-2) progression — parts of value 2⁰, 2¹, 2², …, 2ᵏ⁻¹. With k such parts, every integer from 0 to (2ᵏ − 1) can be formed as the sum of a unique subset of them, since each part independently contributes or does not.

Application:

1. Here N = 31, the value to be fully covered.

2. Use the powers of 2 up to 16: 2⁰ = 1, 2¹ = 2, 2² = 4, 2³ = 8, 2⁴ = 16.

3. Their sum is 1 + 2 + 4 + 8 + 16 = 31, using exactly 5 parts.

4. Every subset of {1, 2, 4, 8, 16} sums to a distinct value, so every amount from 1p to 31p can be given as exact change using some combination of these 5 pieces.

Cross-check: Could fewer than 5 parts work? With n parts, each either present or absent in a payment, there are only 2ⁿ possible subset-sums (including the empty one), so at most (2ⁿ − 1) distinct positive values can be represented. For n = 4, that ceiling is 2⁴ − 1 = 15, which is not enough to reach every value up to 31. Only n = 5 gives 2⁵ − 1 = 31, exactly matching the required range, confirming that 5 is the true minimum.

Result: Hence 5 parts are required.