What's the most efficient way of combining books and tools to reduce experience cost?

Question

In Java Edition Minecraft, you can use an anvil to combine an enchanted book with a tool, granting it that enchantment. However, if you want to apply a second book, the cost will increase, and so on with more books. To reduce the effect of this cost increase, you can combine books with other books for a smaller experience cost, and when applied to the tool, it will only cost the amount that one book would.

The question is, then, what's the most efficient way of combining books with tools to minimize experience cost?

Answer 1

Gnembon covers the problem in his video:

Summarizing his findings, there are three main factors

• level of the enchantment (Fortune III will be three times as expensive as Fortune I); if a book or tool has multiple ones, that's just a sum.
• rarity of the enchantment - a multipler of the level; Efficiency counts as 1x, Fortune - 2x, so applying Efficiency IV will cost as much as Fortune II
• base cost - an extra cost added with every operation, doubles with every subsequent one.

That last part is the most prohibitive, meaning applying enchantments to a tool one after another will quickly drive the cost way up. The way to approach it is to combine enchantments in form of a binary tree - merge same cost books in pairs (then potentially merge these pairs), treating the original tool as one of the 'leaves' of the binary tree (same as one level 1 book).

When combining books swap their positions in the anvil and always choose the cheaper option, and prefer to merge items of similar cost.