The Disappearing Bicyclist
1. When the disk is turned, parts of the 13 cyclists recombine to create 12 new cyclists, each of which takes up a bit more space than the corresponding original cyclist. So instead of saying that one of the cyclists has vanished, it would be more accurate to say that all the cyclists have grown a little, similar to what happens when water from 13 glasses is poured into 12 glasses.

2. The areas are circled above in blue.

Bo Peep’s Pen
1. Bo Peep had eight sheep. The carpenter first used 10 posts to build a 1 x 4 pen with an area of 4 square feet. He then built a 2 x 2 pen, also with an area of 4 square feet but using only 8 posts.

2. The next two larger solutions are 24 sheep and 48 sheep. Twenty-four posts make a 6 x 6 pen with an area of 36 square feet, while 26 posts make a 4 x 9 pen with the same area. Forty-eight posts make a 12 x 12 pen with an area of 144, while 50 posts make a 9 x 16 pen of the same area.

3. Sixteen sheep. Sixteen posts make a 4 x 4 pen with an area of 16 square feet, while 20 posts make a 2 x 8 pen of the same area (solution not shown).

Back From the Klondike
The solution requires that you double back. Loyd invented the number-jumping maze. Modern-day designers such as Adrian Fisher and Robert Abbott have taken the idea even further.