Algebra: Euler’s dice
View Sequence overviewAlgebraic rules can determine unknown values, but whether those values correspond to a real object requires further reasoning.
Whole class
Euler’s dice Slides
Each student
Build my die Student sheet
Gallery walk Student sheet
Construction materials such as paper or carboard, glue or sticky tape, scissors
Task
Remind students of the brief from Task 1: A game designer is creating a board game called Fate and Fortune, a game built entirely around luck, superstition, and the roll of a die. They have one problem: they need a set of custom dice that use a set of numbers considered lucky or unlucky across different cultures.
Show Side 27 of Euler’s dice Slides, which lists the instructions and suggested lucky/unlucky numbers. Each group chooses a number from the list.
Ask: Is your die actually possible? Use the rules from the last task to find out what shape your die would need to be.
A group that chooses 3 will find that no polyhedron with only three faces can be fully enclosed (three flat faces always leave an opening that cannot be sealed). Interestingly, it is possible to find values of $V$ and $E$ that satisfy Euler’s rule for $F = 3$. For example, $F = 3$, $V = 3$, $E = 4$ gives $3 + 3 - 4 = 2$, which satisfies the rule. So does $F = 3$, $V = 4$, $E = 5$. None of these can be assembled into a real closed 3D object, which makes this a useful precursor of the necessary/sufficient discussion that follows.
Let the group work through this and reach their conclusions themselves, then use it as a whole class discussion point. Ask: Why not? What’s the minimum number of faces a polyhedron can have? Once the conversation has landed, invite the group to choose a different number and continue.
Ask: We have Euler’s rule. Does that mean any combination of $F$, $V$, and $E$ that satisfies the rule can actually be built?
Then pose a challenge: Can anyone find a combination of $F$, $V$, and $E$ that satisfies Euler’s rule but can’t actually be built as a real 3D shape?
This is genuinely hard to do from scratch. If the class is stuck, offer these scaffolded prompts one at a time:
- Try very small numbers for $F$, $V$, and $E$. What’s the smallest combination you can find that satisfies the rule?
- Here’s a starting point: $F = 3$, $V = 3$. What would $E$ need to be? Students calculate $E = 4$. Can you picture what this object would look like? Three flat faces always leave an opening and the result cannot be fully closed.
- Once students have established that $F = 3$ is impossible, push further: You’ve shown three faces isn’t enough. But what about larger numbers? Try $V = 3$, $E = 5$, $F = 4$. Does it satisfy Euler’s rule? Students verify: $3 + 4 - 5 = 2$. So, does that mean you can build it?
The reason it fails is simple: three vertices always lie in a single flat plane. No matter how many faces you claim the shape has, three vertices can never enclose any volume.
The key idea to land on: both combinations satisfy Euler’s rule but can’t be built for different reasons.
Once students have seen an example that satisfies Euler’s rule but can’t be built, land on the key idea: Euler’s rule tells us a shape is definitely impossible if it fails. But passing the rule doesn’t guarantee it can be built.
Then connect back to the sequence: At the start of Task 1 we asked what dice are possible. Over the last two tasks we’ve built up some useful mathematics: definitions, patterns, algebraic rules, Euler’s rule. Now we can actually have a go at answering it.
On necessary and sufficient conditions
A necessary condition is one that must be true for something to exist or hold. Failing it rules something out entirely. A familiar example: for it to rain, there must be clouds. Clouds are necessary for rain. If there are no clouds, there is no rain. But clouds alone do not guarantee rain. They are necessary but not sufficient.
In this sequence, satisfying Euler’s rule is necessary for a polyhedron to exist. If $F + V - E \neq 2$, the shape is definitely impossible. A sufficient condition guarantees something exists. Euler’s rule is therefore not sufficient. A combination can satisfy the rule and still not correspond to any real polyhedron.
A necessary condition is one that must be true for something to exist or hold. Failing it rules something out entirely. A familiar example: for it to rain, there must be clouds. Clouds are necessary for rain. If there are no clouds, there is no rain. But clouds alone do not guarantee rain. They are necessary but not sufficient.
In this sequence, satisfying Euler’s rule is necessary for a polyhedron to exist. If $F + V - E \neq 2$, the shape is definitely impossible. A sufficient condition guarantees something exists. Euler’s rule is therefore not sufficient. A combination can satisfy the rule and still not correspond to any real polyhedron.
Hand out the Build my die Student sheet. Tell students: Before anyone starts building, you need to work out what shape your die actually is and check that it’s mathematically possible.
Give groups time to work through the planning organiser before any building begins.
The organiser asks students to:
- choose whether their die will be a prism, a pyramid, or another polyhedron.
- work out the values of $F$, $V$, and $E$ for their chosen object.
- verify using Euler’s rule that their combination is not ruled out.
- sketch the shape and label at least one face, one edge, and one vertex.
- write their pitch to the game designer.
How students find their $F$, $V$, and $E$ values depends on what kind of shape they are designing.
Designing prisms or pyramids
For students designing a prism or pyramid, the rules from Task 2 do the work. The key move is to get students to commit to a shape family first, then work through the rules in order.
Start by asking: Have you decided if it’s a prism or a pyramid? Once they’ve committed, ask: Which rule tells you $B$ from $F$? Students should reach for $F = B + 2$ (prism) or $F = B + 1$ (pyramid) and rearrange to find $B$. Then ask: Now that you know $B$, which rules tell you $E$ and $V$? From there, $E$ and $V$ follow directly.
If students are stuck on rearranging, prompt with a specific example: Your die has eight faces and you’ve decided it’s a prism. $F = B + 2$, and $F = 8$. So, what does $B$ have to be? Once they have $B$, the remaining rules are straightforward substitution.
If students reach for Euler’s rule before they’ve found $V$ and $E$, redirect them: Euler’s rule needs all three values. Which rules from last lesson let you find $V$ and $E$ first? Euler’s rule should be the final check, not the starting point.
Building general polyhedra
For students designing a general polyhedron that is neither a prism nor a pyramid, there are no shortcut algebraic rules. The best starting point is a sketch. Ask: Draw what you think your shape looks like. Now count the faces, edges, and vertices in your drawing. Do those values satisfy Euler’s rule?
The sketch is a first-pass guess and students should expect to revise it. Counting edges accurately from a 2D sketch of a 3D shape can be tricky, and students may find that starting to construct early gives them better feedback than staring at a drawing. A half-assembled paper model makes it much easier to count edges and vertices accurately and to see whether the shape is heading somewhere sensible.
Encourage these students to treat construction as part of the thinking, not something that comes after the thinking is finished. If the model isn’t closing the way they expected, ask: What does that tell you about your sketch? What needs to change?
Designing a general polyhedron is genuinely harder and more open than designing a prism or pyramid. It is appropriate for students who want a challenge, but should not be the default path for groups who are unsure where to start.
Once every group has completed steps 1-3 from the organiser, building begins.
Ask students to complete the first two questions on the Gallery walk Student sheet independently.
Students place their finished die and planning organiser on their desk. Students move around the room, stopping at other desks to examine each die and complete the remaining questions on the sheet.
The completed Gallery walk Student Sheet and Build my die Student sheet together provide a written record of each student’s reasoning across the task and can be used as a summative assessment if needed.
Gallery walks
A gallery walk is a structured activity in which students move around the room to examine and respond to each other’s work. Unlike a class discussion, it gives every student time to engage with multiple responses at their own pace. The purpose is critical comparison, not appreciation. Students are looking at how others approached the same problem, not admiring the finished product.
Verifying someone else’s die against Euler’s rule is a different cognitive task from verifying your own. When students check their own work, they already know what the answer should be. Checking someone else’s die means applying the rule without that scaffolding. Similarly, identifying shape families in an unfamiliar object requires students to use their definitions actively rather than just recognise the shape they built themselves. Some face counts can also be achieved in more than one way. For example, an 8-faced die could be a hexagonal pyramid or a triangular prism. When students notice this, it shifts the question from “what is the answer?” to “how many answers are there?”.
A gallery walk is a structured activity in which students move around the room to examine and respond to each other’s work. Unlike a class discussion, it gives every student time to engage with multiple responses at their own pace. The purpose is critical comparison, not appreciation. Students are looking at how others approached the same problem, not admiring the finished product.
Verifying someone else’s die against Euler’s rule is a different cognitive task from verifying your own. When students check their own work, they already know what the answer should be. Checking someone else’s die means applying the rule without that scaffolding. Similarly, identifying shape families in an unfamiliar object requires students to use their definitions actively rather than just recognise the shape they built themselves. Some face counts can also be achieved in more than one way. For example, an 8-faced die could be a hexagonal pyramid or a triangular prism. When students notice this, it shifts the question from “what is the answer?” to “how many answers are there?”.
Ask students to find the conjecture they wrote at the start of Task 1 (their initial answer to the question “can you make a die with any number of faces?”). Ask: How does your conjecture hold up now? What would you keep? What would you change?
Work through the reasoning together: a pyramid with an $n$-sided base always has $n + 1$ faces, and a prism with an $n$-sided base always has $n + 2$ faces. Since $n$ can be any integer from 3 upward (i.e. the smallest valid polygon is a triangle), a pyramid can have 4, 5, 6, 7... faces, and a prism can have 5, 6, 7, 8... faces. Between the two families, every integer from 4 upward appears. So, any die with four or more faces is mathematically possible. The only impossible face counts are 1, 2, and 3.
The answer to “what dice can we make?” turns out to be almost all of them. The only impossible face counts are 1, 2, and 3. But does 'mathematically possible' always mean 'good'?
Persi Diaconis and the mathematics of fair dice
Persi Diaconis is a mathematician and former professional magician whose work sits at the intersection of probability, statistics and the mathematics of randomness. Among other things, he has studied what it actually takes for a die to be fair.
The intuitive answer is that a fair die needs all its faces to be the same size and shape. This is necessary, but it turns out not to be sufficient. Diaconis and his collaborators showed that true fairness requires something stronger: the die must be isohedral, meaning it looks geometrically identical from every face. More precisely, for any two faces on the die, there must be a symmetry of the shape that maps one face to the other. If this condition holds, every face has exactly the same relationship to the rest of the shape, and therefore the same probability of landing face down.
This is why a hexagonal pyramid fails as a fair die even though all six triangular faces are identical: the base face is fundamentally different from the others in terms of its relationship to the overall shape, and no symmetry can map a triangular face onto the hexagonal base. The triangular faces and the base face are not equivalent under the symmetry of the shape.
The Platonic solids (tetrahedron, cube, octahedron, dodecahedron, icosahedron) are all isohedral, which is why they make fair dice. They are convex isohedral polyhedra with regular faces, which is why the standard set of polyhedral dice contains exactly five shapes.
Persi Diaconis is a mathematician and former professional magician whose work sits at the intersection of probability, statistics and the mathematics of randomness. Among other things, he has studied what it actually takes for a die to be fair.
The intuitive answer is that a fair die needs all its faces to be the same size and shape. This is necessary, but it turns out not to be sufficient. Diaconis and his collaborators showed that true fairness requires something stronger: the die must be isohedral, meaning it looks geometrically identical from every face. More precisely, for any two faces on the die, there must be a symmetry of the shape that maps one face to the other. If this condition holds, every face has exactly the same relationship to the rest of the shape, and therefore the same probability of landing face down.
This is why a hexagonal pyramid fails as a fair die even though all six triangular faces are identical: the base face is fundamentally different from the others in terms of its relationship to the overall shape, and no symmetry can map a triangular face onto the hexagonal base. The triangular faces and the base face are not equivalent under the symmetry of the shape.
The Platonic solids (tetrahedron, cube, octahedron, dodecahedron, icosahedron) are all isohedral, which is why they make fair dice. They are convex isohedral polyhedra with regular faces, which is why the standard set of polyhedral dice contains exactly five shapes.
Famous unusual dice
Two famous examples of unfair dice are worth knowing about for curious students.
The d100 (Zocchihedron): Lou Zocchi’s 100-faced die looks impressive but is not mathematically fair. The vertices are not all identical: in some parts of the shape, five faces meet at a vertex; in other parts, only three faces meet. This means the shape is more “pointy” in some places and flatter in others, which affects how the die sits and rolls. It is a wonderful example of a shape that feels like it should work but fails the deeper geometric conditions.
Fair dice and unsolved problems: Whether a fair 7-faced die is possible is a genuinely interesting mathematical question. A fair die requires all faces to be congruent and the shape to be face-transitive. For 7 faces, no such convex polyhedron is known. The question of which numbers of faces can produce a fair die is still an active area of mathematical research, which can be exciting to share with curious students.
Two famous examples of unfair dice are worth knowing about for curious students.
The d100 (Zocchihedron): Lou Zocchi’s 100-faced die looks impressive but is not mathematically fair. The vertices are not all identical: in some parts of the shape, five faces meet at a vertex; in other parts, only three faces meet. This means the shape is more “pointy” in some places and flatter in others, which affects how the die sits and rolls. It is a wonderful example of a shape that feels like it should work but fails the deeper geometric conditions.
Fair dice and unsolved problems: Whether a fair 7-faced die is possible is a genuinely interesting mathematical question. A fair die requires all faces to be congruent and the shape to be face-transitive. For 7 faces, no such convex polyhedron is known. The question of which numbers of faces can produce a fair die is still an active area of mathematical research, which can be exciting to share with curious students.