Nontransitive dice match
Choose between numerical dice distributions whose pairwise win chances cycle.
Logic & rule checking · Observation & recall
- Players
- 2 players
- Time
- 10–15 minutes
- Materials and quantities
- One ordinary fair die, three face-lookup cards and a two-bout tally sheet.
Each pair keeps two ten-comparison logs; rotate chooser roles, not only the physical die.
Prepare
Use a normal fair die and map its face 1–6 through one of three cards: A gives 2,2,4,4,9,9; B gives 1,1,6,6,8,8; C gives 3,3,5,5,7,7. These are value distributions, not instructions to make or buy special dice. Record the face rolled and the resulting mapped value. Both players roll independently, using the same physical die in sequence if necessary; never reuse one roll as both players’ face.
Play
For a bout, the first chooser selects A, B or C and the second chooser picks a different card. Keep those cards for ten comparisons. Each comparison uses one new die roll per player; the higher mapped value earns one point. The sets have no overlapping values, so comparisons do not tie. After ten, compare points; five each is a drawn bout. Swap first-chooser roles and play one more bout.
Worked example
There are thirty-six equally likely ordered face pairs for two fair rolls. A beats B in twenty, B beats C in twenty and C beats A in twenty: each favourable chance is 20/36, or 5/9. For example A’s face 1 maps to 2 and beats B’s face 1 mapping to 1, but loses to B’s face 3 mapping to 6. The tally sheet separates raw die faces from mapped values to make this auditable.
Finish and check
Finish after two bouts and report both scores and chooser roles; optionally compare total points, keeping ties. A 5/9 edge is not a guarantee of winning ten comparisons. The host matrix counts every face pair, including repeated printed values as separate equally likely faces. Do not compare only the three distinct values as though each had probability one third unless the multiplicities are explicitly maintained.
Adaptation and practical limits
An adult can read the lookup card. Keep the purpose on comparing distributions and observed outcomes, with no money or stakes. If a die is unavailable, use six identical large folded numbered slips, draw one, replace it and mix again before each draw; this is an approximate practical randomizer, not a guarantee of identical physical fairness. The printed key is available after players have explored their choices.
Aim: Choose between numerical dice distributions whose pairwise win chances cycle.
Practice within this activity: Distinguish a probability advantage from a guaranteed outcome.
Adult observation: Audit a mapped value and a counted face pair; do not grade luck.
Mechanism and adaptation: A separately written PartyPageKit version of the mechanism described below, with original examples and materials. The earliest inventor is not established here.
Goals describe observable activity choices, not measured long-term gains or institutional endorsement.
Free alternatives: Use paper or clean reused cardboard and a pencil. Copy the supplied diagrams and tables by hand; no specialist kit or purchase is needed.
Printable materials
Preview at 100% scale, with browser headers and footers off. Cut with suitable help. Keep host keys separate. This is a manual tabletop kit, not an online opponent, automatic judge or solver.
Player die-face lookups
One new independent roll each. Raw face is a column, mapped value is the cell. First chooser then counterchooser; two bouts, ten comparisons each.
| Card / face | 1 | 2 | 3 | 4 | 5 | 6 |
|---|---|---|---|---|---|---|
| A | 2 | 2 | 4 | 4 | 9 | 9 |
| B | 1 | 1 | 6 | 6 | 8 | 8 |
| C | 3 | 3 | 5 | 5 | 7 | 7 |
Player two-bout audit
Bout 1
| Roll | My face/value | Other face/value | Point |
|---|---|---|---|
| ________ | ________ | ________ | ________ |
| ________ | ________ | ________ | ________ |
| ________ | ________ | ________ | ________ |
| ________ | ________ | ________ | ________ |
| ________ | ________ | ________ | ________ |
| ________ | ________ | ________ | ________ |
| ________ | ________ | ________ | ________ |
| ________ | ________ | ________ | ________ |
| ________ | ________ | ________ | ________ |
| ________ | ________ | ________ | ________ |
Bout 2
| Roll | My face/value | Other face/value | Point |
|---|---|---|---|
| ________ | ________ | ________ | ________ |
| ________ | ________ | ________ | ________ |
| ________ | ________ | ________ | ________ |
| ________ | ________ | ________ | ________ |
| ________ | ________ | ________ | ________ |
| ________ | ________ | ________ | ________ |
| ________ | ________ | ________ | ________ |
| ________ | ________ | ________ | ________ |
| ________ | ________ | ________ | ________ |
| ________ | ________ | ________ | ________ |
Host / teacher checked example · separate
Each pair keeps two ten-comparison logs; rotate chooser roles, not only the physical die.
One ordinary fair die, three face-lookup cards and a two-bout tally sheet.
Keep this answer page separate. Read the full web rules and preview materials before leading; allow pauses, passing and supported writing.
A > B
| 1 | 2 | 3 | 4 | 5 | 6 | |
|---|---|---|---|---|---|---|
| 1 | 1 | 1 | 0 | 0 | 0 | 0 |
| 2 | 1 | 1 | 0 | 0 | 0 | 0 |
| 3 | 1 | 1 | 0 | 0 | 0 | 0 |
| 4 | 1 | 1 | 0 | 0 | 0 | 0 |
| 5 | 1 | 1 | 1 | 1 | 1 | 1 |
| 6 | 1 | 1 | 1 | 1 | 1 | 1 |
B > C
| 1 | 2 | 3 | 4 | 5 | 6 | |
|---|---|---|---|---|---|---|
| 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 2 | 0 | 0 | 0 | 0 | 0 | 0 |
| 3 | 1 | 1 | 1 | 1 | 0 | 0 |
| 4 | 1 | 1 | 1 | 1 | 0 | 0 |
| 5 | 1 | 1 | 1 | 1 | 1 | 1 |
| 6 | 1 | 1 | 1 | 1 | 1 | 1 |
C > A
| 1 | 2 | 3 | 4 | 5 | 6 | |
|---|---|---|---|---|---|---|
| 1 | 1 | 1 | 0 | 0 | 0 | 0 |
| 2 | 1 | 1 | 0 | 0 | 0 | 0 |
| 3 | 1 | 1 | 1 | 1 | 0 | 0 |
| 4 | 1 | 1 | 1 | 1 | 0 | 0 |
| 5 | 1 | 1 | 1 | 1 | 0 | 0 |
| 6 | 1 | 1 | 1 | 1 | 0 | 0 |
Rows and columns are raw faces. 1 means left card wins. Each matrix has20 of36 wins =5/9; finite ten-roll bouts can still lose.
A higher mean need not win more comparisons · separate player exercise
| Raw face | 1 | 2 | 3 | 4 | 5 | 6 |
|---|---|---|---|---|---|---|
| D | 0 | 0 | 4 | 4 | 4 | 4 |
| E | 2 | 2 | 2 | 2 | 2 | 8 |
| Comparison | D value | E value | Winner |
|---|---|---|---|
| ________ | ________ | ________ | ________ |
| ________ | ________ | ________ | ________ |
| ________ | ________ | ________ | ________ |
| ________ | ________ | ________ | ________ |
| ________ | ________ | ________ | ________ |
| ________ | ________ | ________ | ________ |
A higher mean need not win more comparisons · host check
D mean=16/6=8/3; E mean=18/6=3. D nevertheless wins4×5=20 of36 independent ordered face pairs; E wins16, with no ties. Repeated numerical faces are separate outcomes, not one distinct-value event.
Related activities
A higher mean need not win more comparisons
Keep the original A/B/C cycle unchanged. This independent two-card practice uses D=0,0,4,4,4,4 and E=2,2,2,2,2,8, with two independent ordinary die rolls. Compare mean value with the proportion of winning face pairs; neither is a promise about a short observed bout.