A computer game consists of a screen in the form of a 3×7 board on which there are three sliding blocks 1, 2, and 3, occupying squares 1×1. The game starts as shown in the figure below, and each move consists of choosing a block and "pushing" it along a row or column. After being pushed, a block will stop only when it encounters the edge of the board or another block. For example, if we choose block 3, we can send it to the bottom right corner or upwards to meet block 2. Two blocks cannot occupy the same square, and when two blocks collide, they do not continue moving. The goal is to make one of the blocks stop on the marked square in the center of the board. Show how this can be done.
!
This one wants a proof. Work it on paper, read the official solution, then mark
yourself honestly — the ladder only means something if the record is true.
Official solution
Solution
Note that to place a block in the marked cell, it is necessary to place another block that prevents the continuation of the slide after passing through that cell. To obtain a solution, move block 2 in the direction of block 1, and then move block 3 to be next to block 2 with two moves. Notice that we can move block 1 to position it to the left of block 3 by moving it down, to the left, up, and finally to the right. Similarly, we can move block 2 to position it to the left of block 1. To finish, we move block 1 down, move 3 to the left, and finally down to reach the marked cell. The figure below illustrates part of the steps performed. ! ! !
#
Source: NuminaMath-1.5,
licensed Apache-2.0.
Statement and solution reproduced as published; topic, difficulty and ordering added
by this site.