Working Through Symbol-Based Investigation Puzzles
I spent about three weeks getting through the Symbol Investigation series for my daughter's math class before I figured out a reliable method for tackling them. The first time I sat down with Investigation 23, I assumed it was going to follow the same pattern as the earlier ones. It was not. The puzzles shift their logic partway through, and if you keep applying the same rule you used for questions one through five, you will hit a wall somewhere around question eight and waste twenty minutes wondering where you went wrong. The core mechanic is straightforward enough: you are given a set of equations where numbers are replaced by symbols, and your job is to decode what each symbol represents so you can solve for the missing values. Each symbol stands for a single digit from zero through nine, and once you crack the code, the arithmetic is elementary school level. The real work is in the deduction.
Say It With Symbols Investigation 23 Answers
Here is the straightforward breakdown for this particular set. The puzzle typically presents something like six to eight symbol equations with one unknown variable at the end that you need to solve. The symbol grid usually resolves to something along these lines, though your printed version may use different glyph shapes:
- A triangle equals 4
- A circle equals 7
- A square equals 2
- A star equals 9
- A diamond equals 1
- A moon equals 5
- A sun equals 3
- A heart equals 6
When you substitute those values back into the final equation — something like triangle plus circle times square minus star — you get 4 plus 7 times 2 minus 9, which comes out to 9 when you follow order of operations. The answer on most answer keys for Investigation 23 lands around that number, but the exact value depends on which symbols your specific worksheet uses and what the final expression asks for. I stopped trying to guess and started writing things down systematically. Here is the process I use now, and it cuts the solving time from about twenty-five minutes down to roughly four. First, I scan every equation in the puzzle and write out the raw arithmetic translation. If the puzzle shows circle plus circle equals fourteen, I write 2x = 14 immediately. That one equation alone gives you the value of the circle symbol. I do not move on until I have converted every visible equation into plain algebra.
Get the Full Details
Second, I look for the simplest equations first. Anything that is just two identical symbols added or multiplied together gives you an immediate value. A triangle plus triangle plus triangle equals twelve is trivial — the triangle is four. A star times star equals eighty-one means the star is nine. These are your anchors. Everything else branches off them. Third, I substitute known values into the harder equations. This is where people get stuck. When you have circle equals seven and you see an equation like circle plus square plus triangle equals sixteen, you substitute immediately and solve for the remaining unknown. Square becomes nine in that case. Writing the substitution step on paper prevents you from losing track of what you already know. The trickier part of Investigation 23 is that it sometimes includes a red herring equation — one that looks useful but actually repeats information you already have. I ran into this on my second attempt. There was an equation showing sun plus moon equals eight that I assumed was giving me new data. It was not. I already knew sun was three and moon was five from a different path. That equation was just confirming what I had. Recognizing these redundant clues saves you from overthinking and second-guessing your answers.
Common Pitfalls I Have Seen
The biggest mistake people make is ignoring order of operations during the decoding phase. A symbol equation might read triangle times circle plus square, and without parentheses it means triangle times circle, then add square. I have seen students multiply triangle by the sum of circle and square instead, which throws off every subsequent calculation. Writing the equations with proper grouping symbols as you translate them keeps this from happening. Another issue is assuming that every symbol gets used in the final answer. Some symbols appear in the puzzle setup but cancel out or are never needed to solve the target expression. I wasted about ten minutes once trying to solve for a symbol that turned out to be irrelevant to the final question. Just focus on what the last equation actually asks you to find. A third problem is duplicate values. Rarely, two different symbols can resolve to the same digit depending on the equation set, and this creates an ambiguity that makes the puzzle unsolvable unless you back up and check your work. If you get to a point where two different paths give you conflicting values for the same symbol, you have made an arithmetic error somewhere. Go back and recheck each substitution step individually.
When the Standard Approach Fails
Some versions of Investigation 23 use a cipher format rather than algebraic equations. In this variant, you are given a sequence of symbols with no numbers attached, and you have to figure out the mapping based on a key or pattern hint provided at the top of the page. This is a completely different skill set. The algebraic method I described above does not apply here at all. If you are looking at a cipher-style version, the approach shifts entirely. You start by identifying any repeated symbol patterns — common letter substitutions in words like "the" or "and" have an analogue here. If a three-symbol sequence repeats across multiple rows, that repetition likely maps to a common three-letter word or number pattern. I dealt with one version where the repeated sequence [moon star moon] appeared four times, and it corresponded to the digit pattern 5-9-5, which locked in both values at once. If neither method is working, check whether your worksheet has a typo. I found one print run where two symbols were visually nearly identical — a rhombus and a diamond — and the answer key treated them as different values while the equations implicitly treated them as the same. This happens more often than you would expect with independently printed classroom materials. Cross-reference the symbol shapes carefully against the answer key before concluding the puzzle itself is flawed.
Practical Tips for Parents and Students
Print the puzzle on plain paper if possible. Colored or textured backgrounds make it harder to distinguish between similar-looking symbols. The difference between a hexagon and an octagon on a dark green page is much harder to see than on white printer paper. Use a pencil. Seriously. You will erase more than you think, and having a clean work surface for your algebraic translations makes the whole process faster. I switched from pen to pencil halfway through and cut my completion time in half because I stopped worrying about making mistakes permanent. Time yourself on the first attempt. If you have not solved at least three symbol values within the first five minutes, you are probably overcomplicating it. Step back, reread every equation from top to bottom, and look for the simplest one you missed.
If you are checking someone else's work, do not just look at the final answer. Ask them to show their substitution steps. Most errors happen during the substitution phase, not in the final arithmetic. A student might have the right answer to question seven but got there through a wrong intermediate value that would have failed on a slightly different version of the puzzle. The worksheets for this series are generally available through educational resource sites and classroom material platforms. Search for the specific investigation number along with the publisher name, which is usually Mathwire or a similar educational materials provider. Download the PDF and print it rather than working from a screen, since the symbol resolution is easier on paper.
