Maths Olympiad Prep

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Problem 540

AMC 10/12, early questions
Combinatorics Difficulty 3.5 Multiple choice CEMC Pascal · Canada · 2021

A robot is placed on the grid shown.

$12345678910111213141516171819202122232425262728293031323334353637383940414243444546474849\begin{array}{| c | c | c | c | c | c | c|} \hline 1 & 2 & 3 & 4 & 5 & 6 & 7\\ \hline 8 & 9 & 10 & 11 & 12 & 13 & 14\\ \hline 15 & 16 & 17 & 18 & 19 & 20 & 21\\ \hline 22 & 23 & 24 & 25 & 26 & 27 & 28\\ \hline 29 & 30 & 31 & 32 & 33 & 34 & 35\\ \hline 36 & 37 & 38 & 39 & 40 & 41 & 42\\ \hline 43 & 44 & 45 & 46 & 47 & 48 & 49\\ \hline \end{array}$

The robot starts on square 25, initially facing square 32. The robot (i) moves 2 squares forward in the direction that it is facing, (ii) rotates clockwise 90∘90^\circ, and (iii) moves 1 square forward in the new direction. Thus, the robot moves to square 39, then turns to face square 38, then moves to square 38. The robot repeats the sequence of moves (i), (ii), (iii) two more times. Given that the robot never leaves the grid, on which square does it finish?

Pick one

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Official solution

Starting at 38, the robot moves 2 squares forward to 36, then rotates 90∘90^\circ clockwise to face 29 and then moves to 29.

Starting at 29, the robot moves 2 squares forward to 15, then rotates 90∘90^\circ clockwise to face 16 and then moves to 16.

Source: CEMC, University of Waterloo, licensed CC-BY-NC-4.0. Statement reproduced verbatim; metadata (topic, difficulty, ordering) added by this project.