42. NOW YOU SEE IT ... 
On a rectangular card, ten parallel line segments are drawn. They are all equally spaced, and all have the same length. The card is cut diagonally as in the diagram. When the two pieces of card are now slid diagonally, one of the drawn line segments can be made to disappear. What actually happens to it?
Hint 1
Make your own model.
Hint 2
Look carefully at the nine line segments remaining after one has been made to disappear.
Solution
We first make a simple observation. Notice that the total length of the line segments remains the same. If we add the lengths of the segments on each part of the card, the sum of all these lengths is unchanged after the pieces have been moved. Thus the nine segments we obtain after sliding the pieces are longer than the original ten segments, and in fact each new segment has a length which is 10 / 9 the length of an original segment..
But this does not explain why a segment disappears. If we had an arbitrary set of equally spaced segments drawn on the cards, we might well expect there to be eleven segments after the move. So why do we get nine?
Suppose the segment lengths on each part are 0, 1, 2, ..., 8, 9.
In the initial position we have:
0 1 2 3 4 5 6 7 8 9
9 8 7 6 5 4 3 2 1 0
with each column adding to 9, giving a total of 10 x 9 = 90.
After the sliding of the pieces we get
0 1 2 3 4 5 6 7 8 9
9 8 7 6 5 4 3 2 1 0.
There are now eleven columns, but the two outside columns have zero length, and so do not appear. There are now 9 non-zero columns, and the total is 9 x 10 = 90 as before. We see that the segment disappears only when the cut passes through the endpoints of the two outside segments in original position.
Extensions
We can think of this problem as a translation problem, with one piece translated against the other. Can you construct a version of this problem which involves rotation?
Hint 1
Hint 2
Solution
Extensions
