This article is the one I find hardest to write because the subject is genuinely difficult to put into words. Mathematical taste — the felt sense that one solution is more beautiful than another — is the thing Abel is most distinctively trying to develop. And it's the thing school maths almost never explicitly teaches.

Let me try to make it concrete.

The two-solutions move

Pick a Year 8-appropriate competition problem. Here's one:

What is the sum of all the integers from 1 to 100?

A school-maths approach: add them up. 1 + 2 + 3 + ... fairly soon you've made an arithmetic error and given up. Or you remember the formula n(n+1)/2, plug in 100, and get 5050. Correct. Done.

The Gauss approach, allegedly discovered by a young Carl Friedrich Gauss when his teacher set this problem as a punishment exercise: pair the numbers — 1+100, 2+99, 3+98 — fifty pairs, each summing to 101. So 50 × 101 = 5050. Done in fifteen seconds without a formula, just by looking at the structure.

Both solutions are correct. The Gauss approach is what mathematicians call elegant. Abel's job is to make a Year 8 student notice the difference, feel the pull of the elegant version, and start hunting for that kind of move themselves.

The five criteria

Mathematical elegance isn't a single thing. After several hundred logged Abel sessions, the criteria that work for KS3 students cluster into five:

1. Brevity

The elegant solution is shorter. Fewer steps, fewer symbols, fewer calculations. Brevity is the most superficial criterion — sometimes a longer solution is the more elegant one because it reveals more — but for KS3, brevity is a useful first signal.

Brute force: 47 lines of arithmetic. Elegant: 3 lines invoking divisibility. The 3-line version is taste-positive.

2. Generality

The elegant solution would work for a whole family of problems, not just the one in front of you. The Gauss trick works for 1 to n for any n. The formula n(n+1)/2 is the formalised version of the same idea.

Question Abel asks: "Would your method still work if the question were 1 to 1000? 1 to 1,000,000?" If yes — your method has generality. That's taste-positive.

3. Structural insight

The elegant solution explains why the answer is what it is, not just that it is. It reveals the structure of the problem. After you've seen it, you understand the problem deeper.

Brute force: "I added them up." Elegant: "Because the numbers pair off in equal sums." The second one teaches you something about how integers work.

4. Surprise

Sometimes the elegant solution is one where the move is genuinely surprising — it uses a tool from somewhere you wouldn't have looked. A geometry problem solved by symmetry. A counting problem solved by parity. A number theory problem solved by drawing a picture.

The "wait, what?" reaction at the moment of the move. That feeling is taste-positive. KS3 students recognise it instantly when they see it.

5. Reusability

The elegant solution introduces a technique you can use again. Mathematicians collect these like a craftsman collects tools. "The divisor trick", "the parity trick", "the symmetry move", "extreme cases". A solution that's elegant in this way leaves you with a new tool.

Question Abel asks after a problem: "What did this teach you to do next time?" If there's a clear answer — taste-positive.

How Abel teaches this in practice

The mechanics are simpler than the philosophy suggests. Four moves Abel uses, repeatedly.

Move 1: "Can you solve it another way?"

After the student has a correct answer, Abel asks for a second method. This is the most reliably useful move. It forces the student into mental flexibility and surfaces whichever method came first — usually the more procedural one — vs. whichever they reach for second — often the more structural one.

Move 2: Show them the elegant version

If the student's solution is correct but brute-force, Abel sometimes shows the elegant version explicitly. Side-by-side. Names the criteria — "look how much shorter", "look how this generalises". The student doesn't feel marked-down; they feel let in on the secret.

Move 3: "What did this teach you?"

After every problem, the reflective close. Not "did you get it right" but "what tool did this give you for next time". This builds the explicit catalogue of techniques that elegant problem-solving draws on.

Move 4: Celebrate elegance in the wild

When the student finds an elegant solution on their own, Abel makes a big deal of it. Not in a stickers-and-streaks way — in a "what you just did is what real mathematicians do" way. The praise is calibrated, specific, and rare enough to mean something.

Why this matters beyond the competition

Here is the bit I most want parents to internalise. Mathematical taste transfers.

A child who has developed taste in maths recognises the same pattern in other domains:

  • Programming: the elegant solution to a coding problem has the same five criteria. Software engineers talk about it constantly.
  • Engineering: the elegant design uses fewer parts, generalises, reveals structural insight. James Dyson made a career out of it.
  • Scientific writing: the elegant paper makes its argument in three pages where a worse paper takes thirty.
  • Legal reasoning: the elegant argument finds the principle that resolves the case.
  • Debate: the elegant rebuttal turns the opponent's strongest point into your evidence.
  • Chess: the elegant move solves multiple positional problems at once.

A 13-year-old who develops the taste muscle in JMC prep doesn't lose it. They take it into whatever discipline they eventually fall in love with. This is one of the highest-leverage skills KS3 maths can develop, and it's one of the most reliably underweighted in school.

The literature, briefly

I want to flag two books for parents who want to read deeper. G.H. Hardy's A Mathematician's Apology (1940) is the canonical statement of mathematical beauty — short, opinionated, beautifully written, available free online. Paul Lockhart's A Mathematician's Lament (2002) is the more accessible polemic on how school maths kills taste by drilling procedure. If you read one as a parent, read Lockhart. If your child likes ideas, give them Hardy when they're fifteen.

I'd also flag Polya's How to Solve It as the canonical practical text. It's the book most mathematical olympiad mentors quietly assume their students have read. Abel's pedagogy borrows liberally from it.

A correct answer that's not enough

I'll close with a specific moment from the prototype phase that crystallised this design for me.

A student solved a JMC-style probability problem correctly using a 12-line case-by-case analysis. The answer was right. The reasoning was sound. School maths would have stopped there.

Abel responded: "Correct! Lovely working. Now — would you like to see how this problem dissolves in three lines if you use complementary counting? It's the same answer, but the second method is one you can use forever. Want to see?"

The student said yes. Abel showed it. The student stared at the screen for a moment and said "oh."

That moment is the entire reason Abel exists. The correct answer was real; the taste lesson was bigger. KS3 students can absolutely have that moment, repeatedly, if a tutor is committed to giving it to them.

Jason runs aitutors.me. He has a Year 8 child who likes Gauss and a chess board within reach of the desk where this was written. Updated 21 May 2026.