1996 IMO Problems/Problem 1

Problem

We are given a positive integer $r$ and a rectangular board $ABCD$ with dimensions $|AB|=20$ , $|BC|=12$ . The rectangle is divided into a grid of $20 \times 12$ unit squares. The following moves are permitted on the board: one can move from one square to another only if the distance between the centers of the two squares is $\sqrt{r}$ . The task is to find a sequence of moves leading from the square with $A$ as a vertex to the square with $B$ as a vertex.

(a) Show that the task cannot be done if $r$ is divisible by $2$ or $3$ .

(b) Prove that the task is possible when $r=73$ .

(c) Can the task be done when $r=97$ ?

Solution

First we define the rectangular board in the cartesian plane with centers of the unit squares as integer coordinates and the following coordinates for the squares at the corners of $A$ , $B$ , $C$ , $D$ , as follows: $A=(1,1)$ , $B=(20,1)$ , $C=(20,12)$ , $D=(1,12)$

Let $(x_i,y_i)$ be the coordinates of the piece after move $i$ with $(x_0,y_0)=A=(1,1)$ the initial position of the piece.

Let $\Delta x_i = x_i-x_{i-1}$ , $\Delta y_i = y_i-y_{i-1}$

Alternate solutions are always welcome. If you have a different, elegant solution to this problem, please add it to this page.

1996 IMO (Problems) • Resources
Preceded by First Problem	1 • 2 • 3 • 4 • 5 • 6	Followed by Problem 2
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1996 IMO Problems/Problem 1

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