An expert question on queues for P5–P6, with a step-by-step solution, diagrams and the answer. Queue questions are already in the Hua Xia Cup's P1 and P2 first rounds, and the P4 papers add square formations.
The question
60 children stand in a circle, numbered 1 to 60 clockwise.
Starting with number 1, they call out "1, 2, 1, 2, …" in turn, clockwise.
Whoever calls "2" leaves the circle at once.
The rest carry on without stopping, each calling the number after the last one called, until only one child is left.
What is the number of the last child left?
Show the hints (3)
💡Hint 1: Try small circles: with 2, 3, 4, 5, 6, 7, 8 children, who is left? What pattern do you see?
💡Hint 2: When the count is 2, 4, 8, 16, 32, …, each lap halves it and the next lap starts again at number 1, so number 1 is always left.
💡Hint 3: 60 is 28 more than 32. Let 28 children leave first, leaving 32. The 28th to leave is number 56; who calls "1" next?
Worked solution
Have a go first, then read the solution step by step.
1.The largest power of 2 not above 60 is 32.
2.When the count is a power of 2, each lap halves it and the next lap starts with the same child, so the child who calls the first "1" is the one left.
3.The first 28 to leave are numbers 2, 4, …, 56, leaving 60 − 28 = 32.
4.Next to call "1" is number 57. The remaining 32 start counting from him, so he is the last left: number 57.
⚠️Common mistake: since the evens leave, answering 1. When the count isn't a power of 2, a lap ends out of step and the next lap no longer starts at number 1.
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