Act/1D hop and skip

Each student in the class is paired with a partner (or triple). The instructions of the activity are as follows:
 * This game is played on a number line. You begin at zero.
 * You are allowed to move along the number line in two ways:
 * You can "hop" by moving 123 units to the left or to the right.
 * You can "skip" by moving 73 units to the left or to the right.
 * The object of the game is to reach the number 1, by a sequence of hops and skips.

Students are given a few minutes to work on this game, without interruption and without shouting out the answer!

Discussion
After playing the game for a few minutes, students are invited to discuss how they solved it. In particular, the following items should be discussed:
 * Can the game be won?
 * How do "compound moves" (call them "jumps" or "leaps", for example) help to solve the game?
 * What solutions were found? How many hops and how many skips did it take?
 * Is there a unique way to win the game?
 * How can you get from one winning way to another?

Transition to the Euclidean algorithm
Set up a table on the board, with the Euclidean algorithm for 123 and 73:

Of course, when done on paper, it looks more like this:

The "sequence of remainders": 50, 23, 4, 3, 1, are "compound moves" in the game of Hop and Skip. Since 1 is a compound move, it follows that it is possible to get from 0 to 1, using only hops and skips.

A variation
Now, play the same game, but with the following two changes:
 * You can "hop" 91 units to the left or right.
 * You can "skip" 56 units to the left or right.

Where on the number line can you get to?