Algebra Warmups - Cryptograms
This page is an introduction to basic cryptogram problem-solving. For more advanced cryptogram problem-solving strategies, please check out the main cryptograms page.
A cryptogram is a mathematical puzzle, where various symbols are used to represent digits, and a given system has to be true. The most common form is a mathematical equation (such as the example below), but sometimes there can be multiple equations or statements.
This example is from the cryptograms warmup quiz.
\[ \begin{array} {c c c } & 1 & E \\ \times & & E \\ \hline & 9 & E \\ \end{array} \]
What digit in place of \(E\) would make the above multiplication true?
Let's start by testing a random digit in this equation, say 8. Check the long multiplication for yourself:
\[ \begin{array} {c c c } & 1 & 8 \\ \times & & 8 \\ \hline 1 & 4 & 4 \\ \end{array} \]
Nope - 8 didn't work, but look at why not -- we could have predicted that 8 would fail because we know that \(8 \times 8 = 64\), and the 1's place of the desired solution, \(9 E\), isn't 8. So let's consider all of the possible digits and find one where the 1's place of \(X^2 = X \times X\) ends with the digit \(X\):
\[\begin{array}{rrl} 1^2 &= 1 &\rightarrow \text{ so 1 works and could be the answer.}\\
2^2 &= 4 &\rightarrow \text{ so 2 does not work.} \\
3^2 &= 9 &\rightarrow \text{ so 3 does not work.} \\ 4^2 &= 16 &\rightarrow \text{ so 4 does not work.} \\ 5^2 &= 25 &\rightarrow \text{ so 5 works and could be the answer.} \\ 6^2 &= 36 &\rightarrow \text{ so 6 works and could be the answer.} \\ 7^2 &= 49 &\rightarrow \text{ so 7 does not work.} \\ 8^2 &= 64 &\rightarrow \text{ so 8 does not work.}\\
9^2 &= 81 &\rightarrow \text{ so 9 does not work.} \end{array}\]So we only have to check 3 possible values that might be \(E: 1, 5,\) and \(6\). And...
\[ \begin{array} {c c c } & 1 & 6 \\ \times & & 6 \\ \hline & 9 & 6\\ \end{array} \]
6 is the one that works! So \(E = 6\).
Guessing and checking \(E=8\) initially wasn't a waste of time, but it's what led us to realize that instead of wasting time with 9 long-multiplication checks, we can just look at the last digits of the perfect squares, and thereby limit our test-cases to 3.
Mathematicians need to be ready (and are usually eager) to get their hands dirty. Even though we are working with numbers instead of physical tools, chemicals, and dangerously high voltage power sources, there's an element of risk to committing to any experiment. Will 5 work? 17? Answering each small question is a bit of work, and it can require a lot of grit to be wrong 1000 times before you're finally right. But, don't let yourself be afraid to guess and be wrong.
\[ \begin{array} { l l l } & & & X & Y & Z \\ & & & X & Y & Z \\ &+ & & X & Y & Z \\ \hline & & & Z & Z & Z \\ \end{array} \]
If \(X, Y,\) and \(Z\) are distinct digits, then what is the value of \(X \times Y \times Z ?\)
Want to solve more cryptogram puzzles? Try the Cryptograms Warmup Quiz.
Quick tips to become a master at solving cryptograms:
- Don't be afraid to start by simply making a guess or two and checking to see what happens. Even if the numbers you guess don't work, you can learn a lot from watching for why they don't work.
- Ask yourself, "What are the properties of the numbers involved in this question and the relevance of the positions that they're in?"
- If a question is too big to tackle all at once, break it into parts. The answers to small questions are frequently necessary stepping stones to discovering the answers to huge, beautiful questions.
- Read the solution if you get stumped, and take careful note of the strategies used so that you can improve your own skills!
Looking for an extra challenge? There are also harder forms for these cryptogram puzzles and more advanced strategies for working on them. If you're intrigued, check out the main, Brilliant Cryptogram Problem Solving page.
Additional articles on Brilliant that are related to solving cryptograms include:
| Wiki \(\hspace{10mm}\) Article | Practice Quiz | \(\hspace{25mm}\) | Wiki \(\hspace{10mm}\) Article | Practice Quiz | \(\hspace{25mm}\) | Wiki \(\hspace{10mm}\) Article | Practice Quiz |
| \(\hspace{20mm}\) Decimals \(\hspace{5mm}\)(Base 10) |
| Simplifying Algebraic Expressions |
| \(\hspace{20mm}\) Linear Equations |
| ||
| \(\hspace{30mm}\)Order of Operations |
| \(\hspace{20mm}\)Divisibility Rules |
| Alternative Base Systems |
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Connecting Algebra And Number Theory
The really deep mathematical patterns that lead to a lot of the logic relevant to solving cryptograms derives from the same reasoning as the divisibility rules, which are also based on digit-by-digit analysis, carried out in base 10. Ever wonder why 1, 5, and 6 are the only numbers whose units digits, when squared, are equal to the original number? Number Theory, with a focus on understanding the base-10 number system, can explain this. But if you're looking to kick it up another few notches, consider that the patterns used to solve cryptograms in other bases would work differently.