IMC 2015 Q12
12. An ant is on the square marked with a black dot.
The ant moves across an edge from one square to an adjacent square four times and then stops.
How many of the possible finishing squares are black?
- A 0
- B 2
- C 4
- D 6
- E 8
Intermediate Mathematical Challenge 2015, Question 12. © UK Mathematics Trust. Original paper on ukmt.org.uk
Answer: D 6
If the ant moves alternately from white square to black square and from black to white, then it will end on a white square after 4 moves. So it must find a way to move from white to white or from black to black. However, there is only one pair of adjacent black squares and only one of white. To reach that pair of black squares, the ant must move to one side then climb up to one of the pair. That uses up 3 moves, and the fourth must take it to the other black square of that pair.
Thus the two black squares in that pair are possible end points.
If, instead, the ant uses the white pair, it must first move to one side, then climb up to one of the white pair then across to the other square of that pair. That uses 3 moves. The fourth move can then take it to any of the three adjoining black squares. This gives 6 end squares, but these include the two already identified.
So there are just 6 possible end squares which are black.
Solution © UK Mathematics Trust, from the IMC 2015 solutions. Longer versions with investigations: Solutions and investigations (PDF).