June 1, 2011 |  by Solomon Golomb
An anagram is a rearrangement of the letters in one word to form a new word. Mathematicians call such a re-arrangement a permutation, and if the permutation leaves no letter in its original position, they call it a derangement (which is also a synonym for insanity). If you write an anagram below the original word, you have a derangement if only different letters line up.
For example,
S    T    A    P    L    E
P    A    S    T    E    L
is a derangement, whereas
S    T    A    P    L    E
P    L    A    T    E    S
is not because the letter A stayed in the same position.
Words sometimes have many anagrams, but derangements are harder to find. One set of letters can’t have more derangements (rows) than the number of letters (columns).
Your present task will be to find three anagrams for each of the numbered sets of letters below in which each is a derangement of the other two. That is, when written as three rows in a rectangle, no column contains a repeated letter. Proper nouns and uncommon words are allowed. The same set of letters may have more than one solution.
1. A, E, M, S
2. A, E, N, T
3. A, E, P, R
4. E, I, L, V
5. E, I, M, T
6. O, P, S, T
7. A, E, L, S, T
8. A, E, P, R, S
9. E, I, P, R, S, T
10. A, E, G, I, L, N, R, T

An anagram is a rearrangement of the letters in one word to form a new word. Mathematicians call such a re-arrangement a permutation, and if the permutation leaves no letter in its original position, they call it a derangement (which is also a synonym for insanity). If you write an anagram below the original word, you have a derangement if only different letters line up.

For example,

S    T    A    P    L    E

P    A    S    T    E    L

is a derangement, whereas

S    T    A    P    L    E

P    L    A    T    E    S

is not because the letter A stayed in the same position.

Words sometimes have many anagrams, but derangements are harder to find. One set of letters can’t have more derangements (rows) than the number of letters (columns).

Your present task will be to find three anagrams for each of the numbered sets of letters below in which each is a derangement of the other two. That is, when written as three rows in a rectangle, no column contains a repeated letter. Proper nouns and uncommon words are allowed. The same set of letters may have more than one solution.

1. A, E, M, S

2. A, E, N, T

3. A, E, P, R

4. E, I, L, V

5. E, I, M, T

6. O, P, S, T

7. A, E, L, S, T

8. A, E, P, R, S

9. E, I, P, R, S, T

10. A, E, G, I, L, N, R, T

Solutions