Paris · May 29, 1832 · the night before a duel

The Night Before
the Duel

A twenty-year-old French mathematician, Évariste Galois, spent his last night alive frantically writing letters to a friend. He had been challenged to a pistol duel at dawn — and he did not expect to survive. In the margins of his manuscripts he scrawled, “Je n'ai pas le temps” — “I have no time.”

He was right to hurry. He was shot the next morning and died a day later. But the pages he left behind answered a question that had defeated the world's greatest mathematicians for 250 years — and in doing so, invented an entirely new idea: the group.

This interactive guide will walk you through exactly what he saw — no university maths required. Just curiosity.

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Part One · The 250-year-old problem

Can you solve every equation with a formula?

You probably know the quadratic formula. It solves every equation of degree 2 — no guessing, no graphing, just plug in the numbers. Mathematicians call this solving “by radicals”: a recipe using only + − × ÷ and roots (√, ∛, …).

History's scorecard looked like this — click each degree:

The state of play in 1824: degrees 1–4 all had formulas, discovered in a burst of Renaissance brilliance. Then… silence. For over two centuries the greatest minds attacked the degree-5 equation (the quintic) and failed. In 1824 the Norwegian Niels Henrik Abel proved the stunning truth: no general formula for degree 5 can exist.

But Abel's proof said that it was impossible — not why. Some degree-5 equations can be solved by radicals. Which ones? And what makes them different? That is the question Galois set out to answer.

Part Two · Galois's leap

Stop staring at the formula. Look at the symmetry of the roots.

Galois's genius was to change the question completely. Instead of asking “what formula gives me the roots?” he asked:

“What ways can I shuffle the roots among themselves so that every true fact about them stays true?”

Try it yourself. Take the equation x² − 2 = 0. Its two roots are √2 and −√2. Press the button to swap them — and watch the facts underneath:

r₁√2
r₂−√2
r₁ + r₂ = 0 ✓ still true
r₁ · r₂ = −2 ✓ still true
r₁² = 2 ✓ still true

Swaps performed: 0 — every fact with ordinary rational numbers survives the swap.

No equation built from + − × ÷ and rational numbers can tell √2 from −√2. They are indistinguishable twins. Swapping them is a symmetry of the equation — like rotating a snowflake.

Galois's big idea: collect ALL the symmetries of an equation's roots and study that collection as an object in its own right. For a degree-3 equation there are up to 6 possible shuffles; for degree 5, up to 120. The structure of that collection — which he called a group — tells you whether a formula can exist.

But what does “the structure of a collection of shuffles” even mean? Let's find out with the friendliest example of all — a triangle.

Part Three · Meet a group

The six symmetries of a triangle

Imagine an equilateral triangle lying in a cut-out hole of the same shape. In how many ways can you pick it up and put it back so it fits exactly? You can rotate it, and you can flip it over. Counting carefully, there are exactly six ways — no more, no less.

Play with it. The dots are glued to the triangle so you can track where each corner goes. The dashed rings mark “home”.

Current symmetry: e (the “do nothing” symmetry)
Your moves so far: —

The four rules that make this a group

Galois's insight was that these six symmetries aren't just a bag of tricks — together they obey four simple laws. Any collection of things obeying these four laws is a group, whether it's triangle symmetries, shuffled roots, clock times, or Rubik's cube moves. Test each law below:

1 · Closed

Combine any two symmetries and you always get… one of the six. Nothing new ever appears. (Press rotate and flip in any order above — the readout always names one of e, r, r², f, rf, r²f.)

2 · Identity

There's a “do nothing” member, e. Doing anything, then nothing, changes nothing. That's the Reset button.

3 · Inverses

Every move can be perfectly undone by another move in the set. Try it:

Pick a symmetry — we'll perform it, then its inverse. You end exactly where you started.

4 · Associativity

With three moves in a row, bracketing doesn't matter: (a·b)·c = a·(b·c).

Part Four · The whole structure on one page

The Cayley table — the group's fingerprint

What happens when you do one symmetry followed by another? All 36 combinations fit in one grid. Click any cell to watch the triangle perform row then column, and reveal the answer.

0 / 36 discovered

Part Five · Feel it in your hands

Galois's challenge: unscramble the triangle

We'll scramble the triangle with a hidden sequence of rotations and flips. Using only Rotate and Flip, bring every dot home. Hint from the axioms: every scramble can be undone, because every move has an inverse.

Moves used: 0
Scrambled… good luck!

Part Six · Why the quintic falls

From symmetries to the answer

Here is Galois's theorem, translated into plain language. A formula built from + − × ÷ and roots can only ever produce symmetries of a very “tame” kind — roughly, symmetries that behave like clock arithmetic (add 3 hours to 10 o'clock and you get 1 o'clock). So:

An equation is solvable by a formula if and only if its group of root symmetries can be completely disassembled into tame, clock-like pieces.

Click through the degrees to see each group's fate:

For degrees 2, 3 and 4 the symmetry groups disassemble nicely — which is why the Renaissance formulas could exist. But the symmetry group of a general degree-5 equation (called S₅, all 120 shuffles of five roots) contains a core of 60 symmetries, A₅, that is indivisible — you cannot break it into clock-like pieces, ever. One stubborn block jams the whole machine, and no formula can be built.

The wider vista. Galois died never knowing what he had started. Today group theory is the mathematics of symmetry itself: it classifies the crystals in salt and snowflakes, underpins the Standard Model of particle physics, secures the cryptography in your browser, and solves Rubik's cubes. All from a twenty-year-old asking, the night before a duel: what are the symmetries of a number?

His manuscripts were nearly lost. Joseph Liouville finally deciphered and published them in 1846 — fourteen years after the duel. The mathematical world has never been the same.