Real problem-solving in maths for Years 3–13, week by week — with step-by-step solutions and a tutor that builds thinking, not memorising. And, underneath it, fluency practice that never hands out the same exercise twice. Physics coming soon.
A snail is at the bottom of a well that is 10 metres deep. Each day the snail climbs up 3 metres. Each night, while it sleeps, it slides back down 1 metre. The snail starts climbing on Monday morning. On which day does it first reach the top and crawl out?
The tutor
Anyone can hand a child the answer — a worksheet key, a leaderboard, an AI. That teaches nothing. PhysMat's tutor does the opposite: it asks the next good question, and stops one step short of solving it for them.
A real hint ladder — strategy, then a sub-step, then a near-solution — that by design never states the final answer. The child gets unstuck, not carried.
Whether you're at home or in the classroom
Enrichment your child will actually enjoy — not another worksheet grind.
A ready-to-set enrichment programme that adds nothing to your workload.
For tutors
One-to-one teaching runs on preparation, and preparation is the part nobody pays you for. Here the year is already laid out: a weekly set for every year from 3 to 13, each problem carrying its worked method and the mistake it invites, and a fluency ladder underneath for the drilling you would otherwise improvise between sessions.
Set work between lessons and it marks itself. What the child did — strengths, weak spots, the direction they are moving — is waiting when you sit down with the parent, so the hour you charge for is the hour you teach.
Beyond one child
Parents and schools are who we build for. But the reason the national curriculum sets fluency, reasoning and problem solving side by side is larger than one kitchen table or one classroom: a country's engineers, doctors, analysts and teachers all begin as children who either learned to sit with a hard problem — or learned to look up the answer.
A child who stops losing marks to arithmetic they never automated, and who learns that being stuck is the ordinary middle of a problem rather than proof they are “not a maths person”, becomes an adult who can reason for themselves. That is why we would rather build this properly than cheaply, and why the tutor is built to withhold the answer even when handing it over would be easier.
One ladder, age 7 to 18
There are 8 children at a birthday party. Every child shakes hands exactly once with every other child. How many handshakes happen in total — and why isn't it 8 × 8 or 8 × 7?
No one shakes their own hand, and every handshake is counted twice. So 8 × 7 ÷ 2 = 28.
At a conference, every delegate shakes hands once with every other. In total 276 handshakes take place. How many delegates attended?
n(n − 1) / 2 = 276 → n² − n − 552 = 0 → n = 24.
The handshake a 10-year-old counts is the quadratic a 16-year-old solves. One canon of problems, from Year 3 to Year 13.
Tap a year to try a real problem from that level.
A drawer holds 6 red, 8 blue and 10 green socks, all mixed up, in a completely dark room. How many socks must you take to be certain of 3 of the same colour — no matter how unlucky you are?
Imagine the unluckiest draw: 2 of each colour is 6 socks with no triple. One more must complete a colour. So 7 socks.
Not a pile of worksheets
What you just saw is the design, not a coincidence: the handshake a ten-year-old counts is the quadratic a sixteen-year-old solves. The canon is built so the same ideas come back, each time at a level that asks more — 8,745 problems across eleven year groups, every one carrying the method it teaches and the mistake it invites.
Underneath sits the other half of the same structure: 72 skills over 360 levels along the arithmetic-to-algebra spine. Each exercise is generated at the moment it is served, and the level it is generated at is chosen from that child's own answers so far — not from a year plan, not from a fixed worksheet order. Problems worth thinking about on top; the fluency they rest on below. Nothing here is a worksheet found on a Sunday evening.
How a week works
A new set of carefully chosen problems lands each week, matched to your child's year.
They think it through. When they're stuck, the tutor nudges — a question, not an answer.
Then the worked solution, step by step: the method named, and the mistakes to avoid.
The other half
The national curriculum asks for three things: fluency, reasoning, and problem solving. A weekly problem serves the last two. The first needs something else entirely — varied and frequent practice, in the curriculum's own words. So we built that as well, as its own thing rather than a compromise between the two.
Fractions and decimals through percentages, ratio, indices, standard form, expanding and factorising, equations, compound measures — the arithmetic-to-algebra spine of KS2 and KS3.
Each exercise is generated the moment it is served, so the answers cannot be memorised and a second run through a skill is genuinely new practice.
The level rises from the child's own recent answers, not from a fixed weekly plan. No timers, no streaks, no leaderboard — accuracy is the point, not speed against other children.
A child who only drills never meets a problem worth thinking about. A child who only meets beautiful problems keeps losing them to arithmetic they have not automated yet. Both, deliberately, in one place.
Classic problems, chosen not generated
The weekly set is not auto-generated. These are the classic problems mathematicians have loved for generations — chosen because the thinking they ask for is beautiful. The fluency exercises above are generated, deliberately and separately: that is what varied practice needs, and it is not what a problem worth an hour of thought needs.
Tap a family to read one.
A team of mowers had to mow two meadows, the larger twice the size of the smaller. For half a day the whole team worked the larger meadow. Then it split in half: one half finished the larger meadow by evening, the other started the smaller and didn't quite finish. The next day a single mower finished the remaining piece in a full day. How many mowers were on the team?
Measure in mower-days, and a paragraph of prose collapses to one line — 8 mowers. Leo Tolstoy wrote it for fun; a twelve-year-old can crack it.
The proof of a maths product isn't a logo wall. It's whether the maths is any good.