Power Play: Thinking Puzzles

Powers grow very fast. Use the laws of exponents, count choices by multiplying, follow doubling backwards, and find units digits from their cycle.

Key ideas

Compare powers by working out their values, or by writing them with the same base.
When each of several choices is independent, the total number of ways is the product of the choices.
A lock with n dials of d symbols each has dⁿ settings.
Something that doubles every day was half-way the day before it was full.
The units digits of powers repeat in a short cycle, so a huge power needs only its place in the cycle.
Optimising means trying the possible choices and keeping the best one that meets every limit.

Remember

Laws of exponents	aᵐ × aⁿ = aᵐ⁺ⁿ; aᵐ ÷ aⁿ = aᵐ⁻ⁿ; (aᵐ)ⁿ = aᵐⁿ
Outfits	4 shirts × 3 trousers × 2 caps = 24
Doubling from 1	day n: 2ⁿ⁻¹; total for n days: 2ⁿ − 1
Units digits of 7ⁿ	7, 9, 3, 1, then repeat: 7²² ends in 9
4-digit codes, first digit not 0	9 × 10³ = 9,000
Teams of k from n, two who refuse to be together	all teams − teams with both

Watch out

Adding the choices instead of multiplying them.
Multiplying the exponents when multiplying powers: 2³ × 2⁴ = 2⁷, not 2¹².
Halving the number of days for “half full” instead of going back one day.

Worked example

A canteen offers 3 starters, 5 main dishes and 4 desserts. (a) How many different meals of one of each are there? (b) One starter cannot be served with one particular main dish. How many meals are possible now? (c) Without that rule, how many meals are possible if one more dessert is added?

Working: 3 × 5 × 4; remove the 4 meals with the banned pair; 3 × 5 × 5

Answer: (a) 60; (b) 56; (c) 75