Use a Finite Combo in Magic the Gathering to Produce Graham's Number

Question

Produce Graham's Number in Magic the Gathering using the fewest number of cards and without using any infinite combos.

Rules:

1. Producing Graham's number means do anything that requires Graham's number to represent the game state. For example, you could produce Graham's number 2/2 zombie tokens, have Graham's number life, or have a creature with Graham's number in power and/or toughness.

2. You must get to Graham's number exactly. If you overshoot it, you need to have a mechanism to get back down.

3. You cannot use a combo that can be repeated indefinitely (commonly referred to as a "infinite combo"), as defined in CR 726. For example, you can't use Basalt Monolith enchanted by Power Artifact, or Squirrel Nest in combination with Earthcraft as either of these can be repeated an arbitrary number of times. You can use abilities repeatedly, such as the ability on Staff of Domination or Voltaic Construct, as long as there is some in-game limit on the number of times it can be repeated (other than the the rules requirement that you not repeat an optional loop forever). The key text from CR 726 (describing interactions that this challenge disallows) is the following:

   726.1b Occasionally the game gets into a state in which a set of actions could be repeated indefinitely (thus creating a “loop”). In that case, the shortcut rules can be used to determine how many times those actions are repeated without having to actually perform them, and how the loop is broken.

4. If the combination involves any cards that let players "choose a number", such as Menacing Ogre, numbers greater than 256 cannot be chosen.

5. You must get to Graham's number in at most 256 turns. (Note that taking infinite extra turns is prohibited as that would be an infinite combo.)

6. You have 256 mana of any color available to you on each turn without it counting against your card limit. Mana beyond that requires you to add cards to do this.

7. You have a single opponent with 20 life. The opponent takes no actions, and does not lose from drawing from an empty deck (assume that they have a deck made of 300 basic lands). You cannot rely on the opponent making specific choices unless you have an effect that lets you control their choices.

8. You must start with a legacy legal deck. In particular, all cards used must be legal in the Legacy format. Additionally, standard deck building rules are in effect; i.e. no more than four copies of any card can be used unless that card specifies otherwise.

9. All of your cards start in either your hand or library (your choice which for any given card). To get a card into another zone requires an ability that moves it there.

10. The solution that uses the fewest number of cards from it's deck and sideboard to complete this task wins. Using a card means that having that card in your deck or sideboard is necessary to completing the combo. Playing the card is the most common "use" of a card, but using a card also includes using it's activated or triggered abilities from a zone outside of the battlefield (such as Decree of Savagery or Bridge from Below) If you use multiple copies of a card, each copy counts as a separate card. Cards that are in the deck merely to meet the 60 card minimum but otherwise are never used are not counted.

Note: Magic the Gathering is known to be Turing Complete, so I assert that this is both possible and as much a coding challenge as those set in Conway's Game of Life.

Inspired by: https://boardgames.stackexchange.com/q/57658/9999

For those who haven't played Magic the Gathering extensively, you don't need to know the entire rule set (which is quite long), as most of that deals with interactions between players, which isn't present for this challenge. The basic rules, supplemented by the comprehensive rules on spells and abilities (section 6) is probably most of what you need to know. The harder part is finding cards that are helpful (as Magic has over 20,000 cards that have been printed). This is the best card search I know about: https://scryfall.com/advanced. Here are some examples of helpful searches:

• https://scryfall.com/search?q=oracle%3Atwice
• https://scryfall.com/search?q=o%3A%22plus+one%22
• https://scryfall.com/search?q=o%3A%22token+that%27s+a+copy%22

Answer 1

6 cards, not yet legal

(The card Saw in Half will become Legacy legal with the upcoming release of Unfinity.)

This solution is based on a solution by plopfill to a slightly different challenge (fixed in a later post), which exceeds Graham's number in 9 cards in 1 turn without free mana: Black Lotus, Show and Tell, Omniscience, Doubling Season, Temur Ascendancy, Astral Dragon, Mystic Sanctuary, Saw in Half, Dromoka, the Eternal.

Doubling Season, Temur Ascendancy, Astral Dragon, Mystic Sanctuary, Saw in Half, Dungeoneer's Pack

Start by playing Mystic Sanctuary and casting Doubling Season and Temur Ascendancy.

Next, cast Astral Dragon. When it enters the battlefield, it triggers Temur Ascendancy's ability and its own ability. Have Astral Dragon's ability resolve first and produce four copies of Doubling Season (because of Doubling Season).

In response to the "draw a card" ability still on the stack, cast Saw in Half, targeting Astral Dragon. The two tokens this produces are doubled five times to make 64 token copies of Astral Dragon.

The first 62 to resolve of the new Astral Dragons' triggered abilities should target Doubling Season. Each one increases the number of Doubling Seasons from \$n\$ to more than \$2^n\$, and the initial number is \$5 > 4 = 2\uparrow\uparrow2\$, so the number of Doubling Seasons after those 62 abilities resolve is more than \$2\uparrow\uparrow64\$ (Knuth's up-arrow notation).

The last two Astral Dragon abilities should target Mystic Sanctuary. The first of those produces more than \$2\uparrow\uparrow65\$ tapped copies of Mystic Sanctuary, and the second produces the same number of copies, but now untapped. Use one of their abilities to put Saw in Half on top of the library, and then the Temur Ascendancy ability draws it.

Cast Dungeoneer's Pack, and then cast Saw in Half on an Astral Dragon again. Target Doubling Season with all but two of the copies' abilities, increasing the number of Doubling Seasons to more than \$2\uparrow\uparrow(2\uparrow\uparrow65)\$. Use the last two abilities to target Mystic Sanctuary and Dungeoneer's Pack for more than \$2\uparrow\uparrow(2\uparrow\uparrow65)\$ copies of each, and again put Saw in Half on top of the library.

Proceed to the second turn, untapping the Dungeoneer's Packs and redrawing Saw in Half in the draw step.

Cast Saw in Half on an Astral Dragon again (although this time, it doesn't make a big difference to the final result). Similarly to before, increase the number of Doubling Seasons to \$2\uparrow\uparrow(2\uparrow\uparrow(2\uparrow\uparrow65))\$ and put Saw in Half back onto the library with a copy of Mystic Sanctuary.

Now start activating the copies of Dungeoneer's Pack. Each one draws a card, allowing another casting of Saw in Half, but the more important parts are the Treasure tokens and the initiative, which gives ventures into Undercity.

With every five ventures, take the path Secret Entrance → Forge → Arena → Archives → Throne of the Dead Three. The important one is Forge: Put two +1/+1 counters on target creature. Keep Saw in Half in hand and target an Astral Dragon with that, giving it more than \$2^{2\uparrow\uparrow(2\uparrow\uparrow(2\uparrow\uparrow(2\uparrow\uparrow65)))}\$ +1/+1 counters the first time.

With its power thereby increased, the copies produced by casting Saw in Half on it will also have high power, and therefore trigger Temur Ascendancy to draw cards, allowing for more castings of Saw in Half.

• One Astral Dragon with power ≥7 gives rise to many with power ≥4, which give card draws and recastings of Saw in Half, each of which gives a tetration; thus the whole thing gives a pentation.
• One Astral Dragon with power ≥13 gives rise to many with power ≥7, for many pentations, thus a hexation.
• One Astral Dragon with power ≥25 gives rise to many with power ≥13, for many hexations, thus a heptation.
• In general, an Astral Dragon with power \$\ge 2^{n}\times6+1\$ gives the \$n+5\$th hyperoperation.

(Also, whenever the black mana is running out, target a Treasure token with an Astral Dragon's ability to make more Treasure tokens, which are usable because Temur Ascendancy gives them haste.)

Forge is reached again with every five ventures, feeding the Doubling Season count into the hyperoperation index each time. The number produced at the end is more than \$2\rightarrow 3\rightarrow (2\uparrow\uparrow(2\uparrow\uparrow65)) \rightarrow2\$ (Conway chained arrow notation), which exceeds Graham's number.

Finally, there are several options for making exactly Graham's number \$G\$ (it's not entirely clear to me what counts):

• Tap \$G\$ copies of Mystic Sanctuary for \$G\$ blue mana.
• Target a Treasure token with the last Astral Dragon ability to have more than \$G\$ Treasure tokens, then sacrifice all but \$G\$ of them.
• Attack with \$G/3\$ 3/3 copies of Doubling Season to deal \$G\$ damage to the opponent.
• Attack with \$G/3\$ 3/3 copies of Doubling Season and twenty 1/1 copies of Astral Dragon to deal \$G+20\$ damage to the opponent and make the opponent's life total \$-G\$.