Crack the Code
People have devised many kinds of secret codes to have private communications with each other, so that no one else can understand their messages.
Letter Substitution Codes
Many messages involve words. One of the most popular ways to encode a message of words is to substitute a different letter for each letter of the alphabet. If the person who gets your message knows your system for replacing letters, it is easy to figure out your code. But even if that person does not know your system, it may not be too difficult to figure out your message, because of certain special letter combinations or the frequency with which certain letters occur.
A Letter-Number Code
This POW concerns codes for arithmetic problems rather than for word messages. To use such a code, you start with an arithmetic problem such as "35 + 35 = 70". To create a coded version of the problem, you replace each number with a letter, always using the same letter for a particular number. For example, you might replace 3 with A, 5 with D, 7 with O, and 0 with H. If you do this, the addition problem becomes "AD + AD = OH". (In using such a code, you need to be careful to distinguish between the number “0”and the letter “O.”)
Figuring Out the Code
It’s easy to make up such a code, and it’s just as easy to figure out what the coded problem represents if you know the replacement system. What’s more interesting is trying to figure out the code merely by looking at the coded problem. That is, you are shown only the problem written with letters, and you have to figure out what the original arithmetic problem was.
The Rules
Problems like these usually follow certain rules.
- If a letter is used more than once in the same problem, it stands for the same number each time it is used.[/*:m:2dw1n94l]
- Different letters in the same problem always stand for different single-digit numbers.[/*:m:2dw1n94l]
- A letter standing for 0 never starts a number with more than one digit.For example, the final arithmetic problem can’t have a number like “05” (but it can use “507”or “80”or even simply “0”).[/*:m:2dw1n94l]
For some letter problems, it is very easy to reconstruct the original arithmetic problem; for others, it is not too hard; and for still others, it is quite difficult. Sometimes there is no possible answer, and sometimes there are many possible answers.
The Problems
See whether you can crack the codes for these problems based on the rules just listed. If you think there is only one right answer, prove it. If you think there are several possibilities, give them all and prove that there are no others. You will need to keep careful track of how you arrive at your answers.
1. ABB + A = DD
2. SS + EE = SST
3. AB + BC = ADE
4. SEND + MORE = MONEY
Note: This one is definitely harder than the previous
ones.
5. Make up an example of your own that has a unique solution, and prove that the solution is unique.
Write-up
1. Process and Solution: Do a separate write-up for each of Questions 1 through 5, combining the process and solution components for each problem. Here, you must prove that your solutions are the only ones possible. You may find that explaining the process you went through to decipher the code will be part (or perhaps all) of your proof.
2. Evaluation
