With the holiday season approaching, Charles Trigg proposed this festive cryptarithm in The American Mathemematical Monthly¹, edition December 1956.

This sentence contains ten different letters. If each letter is a unique representation of a digit, and each word is a perfect square², what are the four numbers?

The puzzle turned out to have two different solutions. The table below shows how the letters have to be replaced by digits to get these two solutions.

The four perfect squares of the first solution are:

The four perfect squares of the second solution are 34225 7396 81 900. If we would fine-tune the puzzle to MERRY XMAS TO YA ALL, then YA is replaced by 59 in the second solution, which is not a perfect square. However, the first solution remains correct, as it results in the following perfect squares: 27556 3249 81 64 400.

Assignment

Solve a puzzle where the words of a given sentence (str) contain ten different uppercase letters. The words of the sentence are separated by a single space.

A solution of the puzzle is a one-to-one correspondence (a bijection³) between the ten uppercase letters in the sentence and the ten digits. A solution is only valid in case each word of the sentence is a perfect square when all letters of the sentence are replaced by their corresponding digit. We use two different representations for a solution.

The dictionary representation of a solution is a dictionary (dict) that maps ten different uppercase letters onto the ten digits, with the digits represented as integers.

The string representation of a solution consists of two strings (str). The first string contains ten different uppercase letters. The second string contains the ten digits, with each uppercase letter of the first string corresponding to the digit at the same position in the second string. The second string does not have to be mentioned explicitly if the ten letters in the first string are mapped onto the digits 0, 1, 2, …, 9.

Your task:

• Write a function solution that takes the string representation of a solution (str; one or two arguments). The function must return the dictionary representation (dict) of the given solution.

• Write a function replace that takes two arguments: i) a string (str) containing the sentence of a puzzle and ii) the dictionary representation (dict) of a solution. The function must return the given sentence (str), where all letters have been replaced by the corresponding digits according to the given solution of the puzzle.

• Write a function issquare that takes a positive integer (int). The function must return a Boolean value (bool) that indicates whether the given integer is a perfect square.

• Write a function iscorrect that takes the sentence of a puzzle (str) and the string representation of a solution (str; one or two arguments). The function must return a Boolean value (bool) that indicates whether the given solution is a correct solution for the given puzzle.

The functions solution and iscorrect must raise an AssertionError with the message invalid solution in case the first argument of the given solution is not a string containing ten different uppercase letters or in case the second argument of the given solution is not a string that contains the ten different digits.

The functions replace and isCorrect must raise an AssertionError with the message invalid puzzle in case the words of the given sentence are not composed of the letters in the given solution or are not separated from each other by a single space.

Example

>>> solution('LOMXARYETS')
{'L': 0, 'O': 1, 'M': 2, 'X': 3, 'A': 4, 'R': 5, 'Y': 6, 'E': 7, 'T': 8, 'S': 9}
>>> solution('AELMORSTXY', '4702159836')
{'L': 0, 'O': 1, 'M': 2, 'X': 3, 'A': 4, 'R': 5, 'Y': 6, 'E': 7, 'T': 8, 'S': 9}
>>> solution('MERYXASTOL', '2765349810')
{'L': 0, 'O': 1, 'M': 2, 'X': 3, 'A': 4, 'Y': 5, 'R': 6, 'E': 7, 'T': 8, 'S': 9}
>>> solution('BANANARAMA')
Traceback (most recent call last):
AssertionError: invalid solution

>>> l2c = solution('LOMXARYETS')
>>> replace('MERRY XMAS TO ALL', l2c)
'27556 3249 81 400'
>>> replace('MERRY XMAS TO YA ALL', l2c)
'27556 3249 81 64 400'

>>> l2c = solution('LORMEYSXTA')
>>> replace('MERRY XMAS TO ALL', l2c)
'34225 7396 81 900'
>>> replace('HAPPY NEWYEAR', l2c)
Traceback (most recent call last):
AssertionError: invalid puzzle

>>> issquare(27556)
True
>>> issquare(34225)
True
>>> issquare(12345)
False

>>> iscorrect('MERRY XMAS TO ALL', 'LOMXARYETS')
True
>>> iscorrect('MERRY XMAS TO ALL', 'LORMEYSXTA')
True
>>> iscorrect('MERRY XMAS TO YA ALL', 'AELMORSTXY', '4702159836')
True
>>> iscorrect('MERRY XMAS TO YA ALL', 'MERYXASTOL', '2765349810')
False
>>> iscorrect('MERRY XMAS TO ALL', 'BANANARAMA')
Traceback (most recent call last):
AssertionError: invalid solution
>>> iscorrect('HAPPY NEWYEAR', 'LOMXARYETS')
Traceback (most recent call last):
AssertionError: invalid puzzle

Resources

• Trigg CW, Ringenberg (1957). E1241. The American Mathematical Monthly 64(6), 432–433. ⁴

• Trigg CW, Thebault V, Rashid MA, Uppal MA, Moser L, Klamkin MS (1956). Problems for Solution: E1241-E1245. The American Mathematical Monthly 63(10), 723–724. ⁵