Alice in Wonderland Puzzle Explained | Beyond Base 10

The “64 = 65” Alice in Wonderland puzzle works because the numbers use different counting bases, not ordinary arithmetic.

Lewis Carroll packed Alice’s Adventures in Wonderland with logic puzzles, wordplay, and mathematical jokes. One of the most famous is the “64 = 65” riddle, which seems to prove that cutting up a square and rearranging the pieces creates extra space. The trick isn’t sleight of hand — it’s a hidden change in the number system itself.

What Is The “64 = 65” Puzzle?

The riddle presents an 8×8 square with 64 unit squares. You cut it into four pieces and rearrange them into a 5×13 rectangle, which should contain 65 unit squares. A chessboard has 64 squares, yet the rearranged shape appears to have 65 — a clear contradiction to ordinary arithmetic.

The illusion works because the pieces don’t fit together perfectly. The edges along the diagonal aren’t a straight line; there’s a thin gap or slight overlap that accounts for the missing square. The visual trick hides this flaw, making the false equality look convincing.

How Does The Base System Solve It?

Modern explanations tie the puzzle to Carroll’s interest in alternative number systems. The equality holds when you interpret the numbers in different bases rather than standard base-10. As one analysis from Maths From The Past explains, 4 × 5 = 12 works in base 18, and 4 × 6 = 13 works in base 21. The same logic applies to the 64 = 65 puzzle: shift the counting system, and the contradiction disappears.

That’s the real trick. Carroll wasn’t making an arithmetic error — he was inviting readers to question what the symbols mean. In one base, 64 really does equal what looks like 65 in another, because the “65” represents a different quantity than it does in decimal.

If you enjoy this kind of clever misdirection, you’ll find plenty more in the Alice in Wonderland puzzle collections worth owning — physical versions of the riddles make great conversation starters.

Why Did Carroll Write This Puzzle?

Carroll was a mathematician who taught at Oxford, and he used the Alice books to explore ideas that seemed playful on the surface but had real mathematical substance. The British Library’s examination of the Alice stories notes that they are full of puzzles, games, and logic challenges woven into the narrative. The 64 = 65 riddle fits that pattern: it’s a genuine brainteaser hiding inside a children’s story.

The TED-Ed lesson on this puzzle also highlights how Alice, the character, refuses to accept claims at face value — she questions, tests, and challenges what the other characters tell her. The riddle works the same way for readers: you have to push past the obvious answer to find what’s actually going on.

That spirit of questioning is the real lesson. The puzzle isn’t just about numbers — it’s about looking closely at what you’re told, checking the assumptions, and refusing to accept a contradiction without investigating it. In that sense, the 64 = 65 puzzle is a miniature version of the whole Alice story.

The puzzle also connects to Carroll’s love of wordplay. Just as a pun works because one sound carries two meanings, the number puzzle works because one symbol carries different values depending on its context. The brain sees “64” and “65” and automatically reads them as ordinary decimal numbers — but the puzzle quietly changes the rules.

Though the false diagonal explains the visual trick, the base-system explanation is the deeper answer. The pieces might not fit perfectly in the drawing, but the mathematical statement “64 = 65” only holds when you stop assuming base 10. Carroll gives you two ways out of the contradiction, and both reward careful attention.