Math Mixer dice puzzle
When I was a young child, my grandparents kept a small collection of toys and board games for me to play with during visits with them. That's where I found a plastic, asterisk-shaped toy. Loosely housed in the middle and at the extremities are five white dice and two black dice. I didn't know what the original intent of the toy was, but for years I had fun rolling the dice and arranging the five numbers of the white dice into a Mathematical expression that equals the sum of the two black dice. Only decades later in 2023 did I finally find it on the internet as a toy called "Math Mixer" with instructions pretty close to what I had developed.
Now as a high school Math teacher, I've used this game for years to promote arithmetic fluency. We play with it throughout quarter 1 of my PreCalculus course, but it can be easily adapted to a wide range of grade levels. I also offer it as an extra-credit puzzle at the end of tests. Students get an extra point for each unique solution they find.
![[color=#b6b6b6]After activating the input bar by clicking/tapping inside it, you may need to click/tap indicated icon in lower left corner to reveal the GeoGebra calculator.
Alternatively, enter your expressions with a standard keyboard.[/color]](https://stage.geogebra.org/resource/uzr55dzj/BMRw3klq2YWk1U3o/material-uzr55dzj.png)
Of course you can play by whatever rules you choose. I typically try first to use ONLY the four basic arithmetic operations (+ – × ÷) applied in whatever order or grouping you like.
But in my classroom, since I'm asking students to get comfortable with more advanced arithmetic operators and to generate as many solutions as possible, here are the constraints we play by.
The following ARE allowed:
- Four basic arithmetic operators (add, subtract, multiply, divide)
- Parentheses, brackets, or other grouping symbols.
- Exponent, using the number on a white die. For example, you may use a 2 from a white die to square another quantity.
- ! (factorial).
- Radical for square root.
- Other roots, using the number on a white die as the index of the radical. For example, you may use a 3 from a white die to take the cube root of another quantity. Admittedly, I don't find many instances for this usage.
- . (decimal point).
- Decimal repeating bar. For example, putting a repeating bar over .3 would make it .333333..., which is equal to 1/3. Note: GeoGebra doesn't have a tool or symbol for a repeating decimal bar. As a workaround, for a repeating decimal, type 'r' after a digit (e.g. ".3r" will be interpreted as ".333333...").
- Using multiple white dice to form a multi-digit number (for example, using 2 and 4 to form 24). Allowing this would make the game too easy in many instances.
- Ignoring a number on a white die. You must use each of the five numbers from the white dice EXACTLY ONCE.
- Utilizing different bases (other than base-10).
- Rounding off/up/down symbols. Cleverly used, such operations would make the game too easy in many instances.
- Use of π, e, or any other Mathematical/scientific constant represented that doesn't show up on the dice.
- More advanced operations not mentioned above, such as logarithms, trigonometry, summations, etc. Permutation/Combination notations and gamma function are about the only "advanced" operations that I could imagine sometimes being useful anyway, and I've always found a way to avoid needing it. (Admittedly rather arbitrary choice to allow factorials but disallow other combinatoric functions).
Example with some solutions
The above example includes 15 solutions, starting with the relatively-simpler ones in "bold" on the left. Usually more solutions than this can be found using only the four basic arithmetic operations. Doing so was difficult in this case because the target 61 is a relatively-large, two-digit prime number.
It's up to the user/class to decide when multiple solutions are too similar to be considered unique. In the example above I expect most would agree that and are essentially the same answer. More subtly though, I would also consider and to be the same. However, I'd consider to be a independent of those two. Other than squabbling over extra credit points, I find such discussions to be a productive part of the pursuit of arithmetic fluency.
Notes on this GeoGebra construction
- In this GeoGebra version, you may use the "max digit" button to toggle between having the maximum number on each die be either 6 and 60 (as in the original toy), or 9 and 90 ('cuz why not?).
- This GeoGebra construction offers a cursory means of validating the user response. A ✔ will display if (1) your response evaluates to the sum of the two black dice, and (2) your response uses all digits on the five white dice exactly once. Otherwise a ✘ is displayed. The construction does not enforce prohibitions such as concatenating digits, using π, using round() or floor() or ceil(), etc. It's not hard to cheat the rules if one chooses, so I leave it up to the user to decide which rules they will observe.
- White dice are automatically greyed out as their respective values appear in the user response.
- Win count is incremented each time a game is won. Only the first correct solution for a given game increments this tally – Subsequent correct solutions will not keep incrementing. Press win count button to reset to 0.
- As mentioned above, enter an "r" to repeat a single digit decimal (e.g. ".3r" will be interpreted as ".333333...").
- To skip the animating spotlight-like circles, click the Roll button a second time during the animation.
- An alternate URL to this page is tinyurl.com/mathmixerdice.
The Original Toy
This GeoGebra version (2023) follows the old HTML5 version of this puzzle (2014) included here for posterity's sake, which itself was converted from its original Adobe Flash format (2007), linked here for a nostalgic chuckle. The original graphics and audio were timeless, so why not stick with them?