The Cycling Problem -and- Crosses & Ticks

In a cycling time trial, cyclists cycle 60 km along the straight road between Linfield and Morton. The first cyclist leaves the starting point at 12:00, the second at 12:04, the third at 12:08, and so on at 4-minute intervals. The cyclist with the quickest overall time is the winner.

Each cyclist cycles at his constant speed throughout the trial.

The 1 cyclist cycles at 15km/h. The 2nd cyclist gains 100 metres on the 1st cyclist during every two minutes.

(a) (i) At what time is the 2nd cyclist level with the 1st cyclist? [2]

At 12:04, 1st cyclist has cycled 1000 m [1]

2nd gains 1000 m in 20 mins, so level at 12:24

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SC: 1 mark for final answer of 12:20 (starts counting at 12:00)

(a) (ii) How far from the starting point are the first two cyclists when they are level? [1]

1st has cycled for 24 mins at 15 km/h,

so distance = 24/60 x 15 = 6 km

(b) What was the constant speed of the 16th cyclist? [3]

2nd cyclist cycles 6 km in 20 mins so he cycles 60 km in 200 mins oe [1]

Finishes at 15:24

16th cyclist leaves at 13:00, so total time is 2 h 24 mins [1]

Speed is 60/2.4 = 25 km / h

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SC: 2 marks for final answer of 25.7… (starts counting at 12:00/13:04)

In a two-player game, players take it in turns, moving left to right, to write either a tick (✔) or a cross (x) in each of the 8 boxes below. Players are not allowed to miss a go.

The rules are:

• two ticks in consecutive boxes must be followed by a cross

• two crosses in consecutive boxes must be followed by a tick

Player 1 begins, and wins the game if they succeed in forcing Player 2 to place a cross in the 8th box. Otherwise, Player 2 wins.

(a) State a valid sequence of ticks and crosses that leads to Player 2 winning. [1]

Any sequence of 8 beginning with C (X) and ending in T (✔) with no run of longer than 2, for example:

CCTTCCTT, CTTCCTCT, CTCTCTCT, etc...

(b) If Player 2's first move is a tick, explain how Player 1 can be certain to win. [2]

​2 marks for a version of 'Player 1 continues with T, C, T'

OR for the sequence (C)TTCCTTC

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1 mark for a version of 'Player 1 continues with T, C'