Magic: The Gathering × Unsolved Mathematics
A combo deck that encodes the Twin Primes Conjecture into a game state — whose maximum damage is infinite if and only if the conjecture is true.
"Mathematics is the queen of the sciences, and number theory is the queen of mathematics." — Gauss
Seven redundant two-card combos. Any one produces unbounded mana. Click "Animate" to watch each loop.
Effigy copies Druid as an artifact creature. Tap for {G}, place −1/−1 counter to untap. As artifact creature, counters don't kill it. Repeat forever.
Druid dies via its own counters. Myrkul returns it as enchantment token (not creature). No toughness = counters meaningless. Tap/untap forever.
Enchant Weaver with Freed. Weaver taps for ≥2 mana (counts Freed + itself). Spend {U} to untap. Net +1 per cycle.
Identical to Combo 3. Pemmin's Aura provides the same {U}: Untap ability. Redundancy doubles assembly odds.
Swift Reconfig makes Druid a non-creature artifact. Keeps abilities, loses toughness. Counters do nothing.
Cast Haze with Buyback. Storm-Kiln creates Treasures per cast+copies. Storm count grows each iteration. Treasures exceed cost exponentially.
Soulbond. Pay {1}{U} to flicker both. Drake ETB untaps 5 lands. Spent 2, got 5. Net +3 per cycle.
Activate for {X}{X}, Sacrifice. Creates X Treasures. Each taps for any color.
{3}{U}: Target player draws a card. With infinite mana, draw every remaining card. Your entire deck is now in hand. The engine needs ~15-20 specific cards simultaneously — Kenrith guarantees access to all of them.
Create Y copies of Mirror Room // Fractured Realms. Unlock each. Result: Y+1 Fractured Realms = Y+1× Zimone trigger multiplier.
Return Mirror Room from graveyard as creature. Unlock → Fractured Realms. Copy ability → new Mirror Room token. Unlock token. Repeat Y times.
Same procedure, different reanimation spell.
No graveyard needed. Turn Fractured Realms into creature via Zur. Use copy ability. Unlock tokens. Repeat Y times.
Sacrificed creatures return as enchantment tokens with all abilities. Feeds on Desecration Elemental's forced sacrifices.
⚠ No alternative existsSacrifice a creature whenever any player casts a spell. Each sacrifice feeds Myrkul, converting creatures to enchantments.
⚠ No alternativeChecks if land count is prime. If yes → creates tokens. Via Life and Limb, every Sapling = Forest = land. Two primes differing by 2 = twin prime = damage.
⚠ Only card in Magic that checks for prime land counts"Tap 3 untapped creatures: Each opponent loses 1 life." Twin prime found → tap creature tokens → 1 damage. ∞ twin primes = ∞ damage.
⚠ No alternative (yet)Tap 2 artifacts → untap target artifact. Untaps Isochron Scepter using artifact lands.
Removes Legend Rule. Zimone creates legendary tokens — without this, duplicates die.
Doubles Zimone's tokens.
Name Sapling. All creatures become Saplings → Life and Limb makes them all lands too.
Every Sapling = Forest land. Every Forest = Sapling creature. Every creature is a land. Every land is a creature. Zimone's land count = total battlefield population.
Grants flash to all spells. Cast sorceries while Zimone triggers are on the stack.
Imprint Narset's Reversal. Cast Molten Duplication targeting artifact land. Scepter casts Reversal → bounces Duplication back, copy resolves → +1 land token. Net: +1 land/cycle.
Imprint Blacksmith's Skill. Target artifact land → Orvar copies it (+1 land). Cast Haze of Rage with Buyback for parity. Net: +1 land/cycle.
If Scepter removed: Elite Arcanist → sacrifice to Desecration Elemental → Myrkul returns as enchantment → Liquimetal Torque makes it artifact → Clock of Omens untaps it.
For sake of argument: Magic allows actual infinity (ℵ₀) as a life total. Kenrith gives opponent infinite life. Engine subtracts 1 per twin prime. Does life reach zero?
ℵ₀ (aleph-null) — countable infinity, the size of {1,2,3,...}. Life totals are naturals, so ℵ₀ is the natural extension. Larger infinities would trivially resolve the question (countable from uncountable always leaves uncountable).
Cardinal subtraction of infinite quantities is indeterminate:
Result depends on which elements removed, not how many. But Magic's sequential subtraction changes everything.
ℵ₀ twin prime pairs. Engine finds them sequentially: pair #n removes life chip #n. This is a bijection. Every chip is removed. Life = 0.
Hilbert's Hotel in reverse: ℵ₀ rooms, each twin prime checks out a guest. ℵ₀ checkouts = empty hotel.
Only K pairs (finite). ℵ₀ − K = ℵ₀. Opponent keeps infinite life. Engine stalls.
Models where you win AND models where you lose. Not covered in Comprehensive Rules.
| Assumption | Twin Primes | Life Remaining | Result |
|---|---|---|---|
| TRUE | ℵ₀ | 0 (bijection) | You win |
| FALSE | K (finite) | ℵ₀ | You lose |
| Independent | ? | ? | Undetermined |
Magic finds twin primes one at a time, in order. This converts ill-defined ℵ₀−ℵ₀ into a well-defined ordinal process. Key theorem: if a countably infinite set has elements removed one-at-a-time indexed by ℕ, and every element is removed at some finite step, then S is empty.
The game's outcome is logically equivalent to the truth value of the Twin Primes Conjecture. A proof of the conjecture = a proof this deck wins.
99 maindeck + 14 sideboard. Hover any card for preview.
This engine asks the Twin Primes Conjecture — in cardboard — with mathematical rigor bridging number theory and a card game.
damage = min( ⌊(Y+1)/2⌋, |{twin prime pairs}| ) where Y → ∞Whether that question has an answer is left as an exercise for the mathematicians.