Guess Like a Mathematician: How Information Theory Wins Guessing Games

A 2026 Binghamton study solved 99% of Wordle puzzles with Shannon entropy. Learn 3 rules — split, probe, budget your bits — to win Bulls and Cows, Nerdle & more.

Every guessing game asks you the same question in disguise: what should you try next? Most players answer with a hope. They guess the word they think is most likely, the number that feels right, the pattern they almost remember. Mathematicians answer differently. They treat every guess as a question whose job is not to be right — it is to find out as much as possible.

That idea comes from information theory, a branch of math born in 1948, and it is not just theory. In 2026, a research team at Binghamton University used it to solve 99% of Wordle puzzles in simulation, while a conventional strategy solved 90% [1]. In this guide we turn that math into three plain rules you can use tonight — on higher-or-lower, on Bulls and Cows, on Nerdle, on any game where feedback tells you how close you are.

Every Guess Is a Question About Possibilities

Start with the reframe. When you type a guess into a code-breaking game, you are not really gambling on that guess being the answer. You are running an experiment. The feedback — green and gray tiles, "bulls and cows" counts, "higher" or "lower" — splits your list of remaining candidates into groups. A great guess splits that list into tiny pieces. A weak guess barely cuts it.

What entropy actually measures

Claude Shannon, a mathematician at Bell Labs, gave this intuition a number. His 1948 paper A Mathematical Theory of Communication — the founding document of information theory, preserved today in IEEE's educational archive [2] — defined entropy as a measure of uncertainty: how many possibilities are still alive, weighted by how likely each one is. Shannon also gave us the bit, the unit of information you earn every time a yes/no question cuts the possibilities in half.

A 1940s mathematician's desk with a branching decision tree drawn on a chalkboard

Here is all the math you need: if there are N possibilities left, you need about log₂(N) well-chosen guesses to guarantee a win. Not because the math is magic, but because each maximally informative guess roughly halves the space. 64 candidates → 6 sharp questions. 1,000 candidates → 10. The log is your information budget, and good players spend it deliberately.

The 99% Experiment: What the 2026 Binghamton Study Found

Does maximizing information actually beat playing it safe? The cleanest test came from Binghamton University (SUNY), where a research team led by Congyu Wu applied Shannon entropy to Wordle [1]. Their algorithm scored every candidate guess by how evenly its possible feedback patterns split the remaining answer list, then picked the most informative option — even when that word had almost no chance of being the answer.

The result: the entropy-maximizing strategy solved 99% of puzzles within six guesses, versus 90% for a conventional strategy built on common letters like A, E, and R. As doctoral researcher Donald Stephens put it in the study's coverage by ScienceDaily, "a guess doesn't have to be the most likely answer; it simply has to be informative" [3].

The counterintuitive part is that the winning guesses often look wrong. When ten candidates remain, the entropy approach may suggest a probe word that eliminates eight of them at once — sacrificing this round's small chance of winning for a much better position next round. Sci.News' report on the study frames the whole game this way: a dynamic feedback system in which entropy (uncertainty) starts high and falls with every informative guess, moving the game "from disordered state... to more organized state" [4].

That is the whole skill in one sentence. Drive the entropy down as fast as possible, and the win takes care of itself.

Blank green, amber and gray tiles arranged in a grid, some set aside

Rule 1 — Split, Don't Stab: Balance Your Guesses

Rule one: prefer guesses whose feedback splits the remaining candidates into roughly equal piles. This is the "divide the search space" instinct, and it is easiest to see in Higher or Lower.

Suppose a hidden number is somewhere in a 100-value range. If you open with the midpoint, every outcome is maximally useful: "higher" leaves 50 candidates, "lower" leaves 49, and a hit ends the game. Open near the top of the range instead, and a "lower" answer leaves you almost the whole space — you spent a guess and learned almost nothing. Balanced splitting guarantees you finish a 1–100 range in seven guesses or fewer, no luck required.

The same logic applies with richer feedback. In Bulls and Cows, a first probe like 0 1 2 3 doesn't need to be right; it needs to produce a "bulls and cows" count that sorts the thousands of possible codes into distinct buckets. A probe whose feedback varies a lot across candidates is informative. A probe whose feedback is the same no matter the answer — like repeating a digit — is wasted information.

The practical habit: before you guess, ask "if the answer is X, what feedback do I get? What if it's Y?" If your honest answer is "pretty much the same either way," pick something else.

Wooden game pieces split into two exactly equal piles by a brass rod

Rule 2 — Spend Early Guesses on Information, Not on Winning

Rule two: early guesses are for buying information; late guesses are for cashing it in.

This is where human intuition fails hardest. In Bulls and Cows Mastermind, players often open with a "sensible-looking" code they believe could be the answer. Better: open with a fixed, balanced probe — four distinct digits spanning the low and high ranges — and treat its feedback as pure data. Then switch to elimination mode: each subsequent guess should test a specific hypothesis about which digits are in the code and where. The same discipline powers 24-point solvers and Mastermind engines, which open with coverage-oriented probes rather than lucky hunches.

You can feel the difference in Nerdle. The opening equation 58-46=12 is popular not because it is ever the answer, but because it touches eight slots of arithmetic at once — subtraction, equality, two-digit numbers, a borrow. One guess, and the color feedback has already narrowed the board enormously. Gamers who open with a random "real-looking" equation are, statistically, buying one lottery ticket instead of a map.

Try it now on Higher or Lower: in any number-based round, the midpoint of the live range is never a bad guess. Boring? Yes. Undefeated? Also yes.

Rule 3 — Count Down the Bits: Know When You're Guaranteed

Rule three: do the budget math, and stop panicking. Once you know how many possibilities remain and how many guesses you have left, you know whether you can afford elegant play or must get aggressive.

A 1–100 range → 7 balanced guesses guarantees a hit (2⁷ = 128).
A 1–1,000 range → 10 guesses (2¹⁰ = 1,024).
A 4-digit Bulls and Cows code with no repeated digits → 5,040 candidates, about 13 bits of uncertainty. Six or seven sharp probes usually corner it — and Donald Knuth proved the related game Mastermind can always be cracked in five guesses with the right strategy.
Nerdle's answer list holds more than 17,000 valid equations — about 15 bits. That sounds brutal until you notice each colorful row of feedback is worth far more than one bit. Well-played, you rarely need the full budget.

The number games on iQiQGame reward exactly this accounting. If you like the arithmetic side, our guide to the 24 Game's strategy framework shows the same idea from the composition side — instead of eliminating candidates, you build combinations forward. Two halves of one skill: knowing the space, and cutting it down.

Vintage abacus with beads slid halfway, beside an open blank notebook

Your 3-Rule Cheat Sheet

Split, don't stab. Choose guesses whose feedback divides the remaining candidates as evenly as possible.
Buy information early. Opening guesses probe coverage; endgame guesses chase the win. Informative beats likely.
Budget your bits. About log₂(candidates) sharp guesses guarantees success — so count, and stay calm.

Seventy-eight years ago Shannon turned "how much do we know?" into arithmetic. The 99% result from Binghamton shows the arithmetic wins. Bring it to your next round — the puzzle doesn't stand a chance.