Counting, Probability & Logic: GK UKMT Intermediate GK 2016 Q20

GK 2016 Q20

GKCurrent spec6 marksCounting, Probability & Logic

20. The cube shown is divided into 64 small cubes. Exactly one of the cubes is grey, as shown in the diagram. Two cubes are said to be ‘neighbours’ if they have a common face.
On the first day, the white neighbours of the grey cube are changed to grey. On the second day, the white neighbours of all the grey cubes are changed to grey.
How many grey cubes are there at the end of the second day?

A 11
B 13
C 15
D 16
E 17

A 4 by 4 by 4 cube with one small cube on the top face shaded grey

Grey Kangaroo 2016, Question 20. © UK Mathematics Trust. Original paper on ukmt.org.uk

Solution (UKMT)

The diagram below shows the day on which certain cubes turned grey.

top layer
2	1	2
1	0	1	2
2	1	2
	2

second layer
	2
2	1	2
	2

third layer

	2

As can be seen, at the end of the second day there are 11 + 5 + 1 = 17 grey cubes.