How is the Minecraft enchantment order worked out?

Minecraft

Enchantment Order

Put enchanted books on an item in the wrong order and the anvil says Too Expensive! before you have finished — or charges twice the levels it needed to. The order matters because every anvil use makes the next one dearer, and the calculator finds the cheapest order there is.

Enchantment Order

Pick the item and the books you have (at max level). This finds the anvil order that costs the fewest levels — Java Edition rules.

Item

Enchanted books

Times the item has already been on an anvil

Total levels

Most expensive step

Limit is 39

Step 13 levels
Left slot: Sword
Right slot: Book: Unbreaking III
Step 22 levels
Left slot: Book: Sharpness V
Right slot: Book: Mending I
Step 39 levels
Left slot: Sword (Unbreaking III)
Right slot: Book: Sharpness V + Mending I

How the numbers are worked out

Each anvil step costs the enchantment levels on the right-hand book, multiplied by that enchantment's book multiplier, plus a penalty for how many times each of the two items has already been through an anvil: 2^n − 1, so 0, 1, 3, 7, 15 levels. The result carries the higher count plus one. Pairing books with each other first keeps penalties low, but the best pairing depends on the multipliers, so the calculator tries every possible merge order and keeps the cheapest one where no single step reaches 40 levels.

The formula

step cost = Σ (level × book multiplier) of the right-slot book + (2^n₁ − 1) + (2^n₂ − 1)
n = anvil uses so far; the result has n = max(n₁, n₂) + 1
a step of 40 or more is Too Expensive!; every merge order is tried and the cheapest total kept

Common mistakes

These are Java Edition rules; Bedrock prices anvil work differently. The total is in levels, and levels are not all equal — going from 30 to 31 takes far more experience than 0 to 1 — so spending levels on the big steps while your level is low is cheaper than it looks. Renaming the item on the anvil also adds a level to that step.

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