When Sam Loyd died in April 1911 at age 70, The New York Times described him as a man with “a real gift . . . for the fantastic in mathematical science.” Loyd, who was also a renowned chess strategist and skilled ventriloquist, created more than 10,000 puzzles during his lifetime. Here are three gems adapted from The Mathematical Puzzles of Sam Loyd (Dover Publications, 1959) and More Mathematical Puzzles of Sam Loyd (Dover Publications, 1960), both edited by Martin Gardner.

The Disappearing Bicyclist

Loyd patented this mechanical puzzle in 1896. The original showed Chinese warriors; this variation, from 1906, features cyclists. Imagine that the bicycle wheels above are cardboard disks that spin atop a cardboard background. The portions of the cyclists that extend beyond the wheels’ edges are printed on the background.

1. [Challenging]  Thirteen cyclists appear in the figure shown above left. If you turn the wheel from A to B, one of the cyclists disappears, as shown at right. Can you explain where the 13th cyclist went?

2. [Easy]  To make the cyclist vanish, Loyd employed clever pictorial trickery. Find part of a fist that is also part of a face and two right legs that are also left legs.

Bo Peep’s Pen

1. [Easy] Bo Peep hired a carpenter to build a pen for her sheep. She intended not only to enclose her sheep but also to tie them to the fence posts. The carpenter first built a long, rectangular pen with posts at one-foot intervals. But then he found he could save two posts by building a square pen with exactly the same area as the rectangular one while keeping the posts at one-foot intervals. Furthermore, each sheep could now be tied to a different post, with no posts left over. How many sheep were in Bo Peep’s flock? Hint: To help you visualize the puzzle, consider the fence surrounding this puzzle’s title (left), which has 18 posts and an area of 20 square feet (4 x 5). The solution has no more than 20 sheep.

2. [Challenging] Find two solutions to Bo Peep’s Pen that allow for more than 20 sheep.

3. [Challenging] In the first problem, suppose the carpenter had found a way to save four posts instead of two. How many sheep would have been in this flock? Again, the solution involves no more than 20 sheep.

Back From the Klondike

This maze was among the many puzzles Loyd created for newspapers beginning in 1890. The numbers in each square in the grid indicate how many squares you can travel in a straight line horizontally, vertically, or diagonally from that square. For example, if you start in the red square in the center, your first move can take you to any one of the three yellow squares.

1. [Easy] Can you travel from the red square in the center to the topmost green square in just two jumps?

2. [Challenging] Can you figure out how to jump from the red square in the center to each of the other three green squares? The trip to one of the squares will take two jumps, the next three jumps, and the last one four jumps.

3. [Difficult] Loyd’s original challenge was to figure out how to move from the red square in the center to one of the purple squares around the edge. The final jump must land on a purple square without passing over any other purple squares. Here’s a big hint: All but the last jump are along a single straight line.