TTMUA Lab
TMUA 2019 · Paper 2 · Question 9 of 20

TMUA 2019 Paper 2 Question 9

Number and divisibility — Covering a cycle · a counting argument. Try it first; the answer and a full worked solution are below.

TMUA 2019 · Paper 2Number and divisibilityCovering a cycle · a counting argument6 options
A large circular table has 40 chairs round it.

What is the smallest number of people who can be sitting at the table already such that the next person to sit down must sit next to someone?

  1. A9
  2. B10
  3. C13
  4. D14
  5. E19
  6. F20
Show the answer and worked solution
answer · D
  1. A9
  2. B10
  3. C13
  4. D14
  5. E19
  6. F20
The condition is that every empty chair has an occupied chair beside it. With k people there are 40 − k empty chairs, and each person can be beside at most 2 of them, so 40 − k ≤ 2k, giving k ≥ 403 and hence k ≥ 14. Thirteen is not enough: thirteen people leave thirteen gaps totalling 27 chairs, so some gap holds at least three empty chairs and its middle chair has two empty neighbours. Fourteen is achievable — seat people with gaps of two empty chairs between them, using one gap of only one chair to make the numbers come out at 40.