Brain-Twisters I

The puzzles in this page will generally be examples of the categories I enjoy most - those involving geometry, topology, probability, and truth paradoxes. Presenting them will often involve the use of tables and figures. The page should still be readable with text-based browsers such as Lynx, provided the reader has some way of downloading and viewing gif figures.

Puzzle #1

This is the first of a series of classic "Lady or the Tiger" puzzles. You have to choose between two rooms. Each of them contains either a lady or a tiger, but it could be that there are ladies in both rooms, or tigers in both rooms, or one could contain a lady while the other contained a tiger. There are signs on the doors of the rooms:

I

IN THIS ROOM THERE IS A LADY, AND IN THE OTHER ROOM THERE IS A TIGER

 II

IN ONE OF THESE ROOMS THERE IS A LADY, AND IN ONE OF THESE ROOMS THERE IS A TIGER

One of the signs is true, but the other one is false. Which door would you open (assuming, of course, that you preferred the lady to the tiger)?

Solution.

Puzzle #2

This time the signs read as follows:

I

AT LEAST ONE OF THESE ROOMS CONTAINS A LADY

 II

A TIGER IS IN THE OTHER ROOM

The statements are either both true or both false. Which room should you pick?

Solution.

Puzzle #3

The signs now read:

I

EITHER A TIGER IS IN THIS ROOM OR A LADY IS IN THE OTHER ROOM

 II

A LADY IS IN THE OTHER ROOM

Once again, the statements are either both true or both false. Does the first room contain a lady or a tiger? What about the other room?

Solution.

Puzzle #4

Langley's Adventitious Angles. This tricky problem, named after E. M. Langley, is famous because it is not as simple as it seems.

Download this figure for Puzzle #4

ABC is an isosceles triangle such that AB = AC and the vertex angle BAC is 20 degrees. Points D and E lie on AC and AB respectively such that angle DBC = 60 degrees and angle ECB = 50 degrees. All you have to do is find angle BDE.

Solution.

Puzzle #5

Byronia, a small planet that orbits a sun near ours, has a humanoid population similar to our own. The most striking difference is that Byronians come in three sexes: male, female, and hermaphrodite. Since hermaphrodites have both male and female organs, they can perform as either sex and also bear children. Whenever a "mother" (female or hermaphrodite) gives birth, the probability that the child is male, female, or hermaphrodite is exactly one-third for each.

The new Supreme Ruler of Byronia, Norman Machismo, is a virile, hot-tempered male who gained total power by defeating a rebellious army of hermaphrodites. To solve the "hermaphrodite problem" Machismo has issued the following decree: Every mother on Byronia, as soon as she or it gives birth to a hermaphrodite, is to be rendered incapable of further conception.

Machismo reasoned like this. Some mothers are sure to have two, three, four, or even more male and female children before giving birth to a hermaphrodite. True, occasionally, a mother will have a hermaphrodite first child, but that will be the end of her childbearing, so these births will contribute only a small percentage of hermaphrodites to the population. In this way the proportion of hermaphrodites in the population will steadily diminish.

Will the Supreme Ruler's plan work?

Solution.

Visit:

Puzzles 1-3 adapted from Smullyan, R. (1982) The Lady or the Tiger? and Other Logic Puzzles, Knopf.
Puzle 4 adapted from Tripp, C. (1975) 'Adventitious Angles', The Mathematical Gazette, Vol 59, No. 408.
Puzzle 5 adapted from Gardner, M. (1981) Science Fiction Puzzle Tales, C. N. Potter.
Revised: July 29, 1996