Bogglers Solutions

Bogglers Solutions

By Scott Kim|Friday, February 01, 2002

Knot a Problem

1. Four elevator rides.

2. Four elevator rides.

3. Six floor changes.

After trying this puzzle you might enjoy finding ways to walk a knotted path in your daily life, or ponder how many overpasses you might need to drive your car in a knotted path. For more information about the mathematics of knots, see Knot Book by Colin C. Adams (W. H. Freeman, 2001).

Hanging Tough


3. To construct the solution for three nails, start by doubling the wire, making a large loop. Then position the two looped ends over the third nail as shown below. If you were to let go at this point, the wire would fall. And if you were to hold the double loop and remove the nail, the two ends of the loop would not stay connected.

As shown above, wrap the doubled wire around the first two nails the same way you did for the solution to the two-nail problem. If you remove either of the first two nails, the wire slips off the other nail, and the remaining loop falls off the third nail. If you remove the third nail, the loop breaks open and falls off the other nails. The wire must cross over and under exactly as shown above.

4. To wrap a wire around four nails so that removing either one of two specific pairs of nails causes the picture to fall, replace each nail in the two-nail solution with two nails as shown above. The only way to make the picture fall is to remove both left or right nails.

To wrap a wire around four nails so that removing one of four specific pairs of nails makes the picture fall, wrap the wire around nails 1 and 2 and nails 3 and 4 as shown above, using the two-nail solution. Removing any one nail will make the wire slip off its partner. To make the picture fall, you must remove either the two outside nails, the two inside nails, or one outside and one inside nail. Removing just the left pair or the right pair will not make the picture fall.

For more about the picture-hanging problem, see

Hold That Line

1. A tetrahedron. The darkened lines show the lengths of string held by each of the four people.

2. An octahedron. Note that the two darkened triangles, one on the top and one on the bottom, are both parallel to the ground and that none of the edges have doubled strings.

3. The most symmetrical way for four people to make a cube with a loop of string starts with the construction of the tetrahedron in problem 1. Once all four people have touched hands at the center, each person passes the middle of the length of string to the hands of the person opposite, to form the center figure above, while continuing to hold on to the original two points. Then everyone pulls apart slightly, widening the top and bottom edges of the tetrahedron until they become squares.

4. Orient the octahedron so that one triangular face is on top and the opposite parallel triangle is on the bottom. To make a cube, the fourth person gathers the midpoints of the three edges of the top triangle in one hand, the midpoints of the three edges of the bottom triangle in the other hand, and pulls his hands apart.

For more information about mathematical string figures and other kinesthetic mathematical activities, head to For more about string figures, see the International String Figure Association at

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