How these are built
The range sets the arithmetic. The level sets the thinking.
Two controls on this builder look as if they do the same job and do not. Numbers up to decides which sums a child has to do: at up to 20, 33% of the sums in a pyramid cross a ten, and at up to 1,000 it is 69%. Difficulty decides what a child has to notice: whether every gap is two numbers added, whether some have to be worked backwards, or whether the sheet can only be finished by calling a missing number n. Either can be turned up without the other.
Each level is named after the step its sheets force
A pyramid says each brick is the two below it added together. A magic square says every row, column and diagonal makes the same total. Underneath, both are a set of sums with numbers missing, so one solver grades both, and it works the way a pupil does: it looks for a sum with a single gap and fills it. There are three kinds of step, and each level is named after the hardest one its sheets cannot be finished without.
| Level | Number pyramids | Magic squares | Numbers printed |
|---|---|---|---|
| Easy | Add up. The whole bottom row is printed. Every other brick is two numbers added together. | Work backwards. The total is printed and every gap is found from a line with one number missing. | 4 of 10 bricks · 3 of 9 squares |
| Medium | Work backwards. Some bricks can only be found backwards, as the brick above minus its neighbour. | Work backwards. No total is printed. Add up the one complete line to find it, then work as on Easy. | 4 of 10 bricks · 4 of 9 squares |
| Hard | Call it n. At some point no brick has a single gap. Call one missing brick n and follow it until a sum says what n is. | Call it n. The total is printed, but no line can be finished on its own. Call one square n and follow it through the lines. | 4 of 10 bricks · 2 of 9 squares |
Read from the engine at build time: 12 sheets per cell, pyramids with a base of 4 and squares 3 × 3, both with numbers up to 100. Every number left on a sheet is one it cannot be finished without; the generator removes the rest.
Why a Medium magic square prints more numbers than an Easy one
Every gap in a magic square is a missing part of a line's total, so a square has no add-only level to start from the way a pyramid does. Medium leaves the total off instead. That forces the sheet to keep one complete line, because a line with every number showing is the only way to find the total, and the result is a sheet with 4 numbers on it that is still harder than an Easy sheet with 3. Counting printed numbers is how most worksheets grade difficulty, and this is the clearest case on the page of why that is the wrong measure.
Numbers up to changes the sums, and squares ask for more carrying
The range is the arithmetic control. It caps every number on the finished sheet, answers and totals included, so a sheet set to 20 never asks a child to write 21 anywhere. What it changes is how often a sum crosses a ten, which is where most mistakes in written addition come from.
A wide pyramid cannot stay under 20
The top brick of a pyramid is its base added up with weights, and the weights are a row of Pascal's triangle. With a base of 5, the middle base brick is counted 6 times in the top and each corner once, so a base of all ones already gives a top of 16. Size and range are separate controls, but they are not independent.
| Base | Smallest top | Up to 10 | Up to 20 | Up to 100 | Up to 1,000 |
|---|---|---|---|---|---|
| 3 | 4 | 50 | 525 | 79,625 | 83 million |
| 4 | 8 | 6 | 330 | 401,016 | 5 billion |
| 5 | 16 | None | 17 | 583,780 | 84 billion |
| 6 | 32 | None | None | 193,410 | 506 billion |
Different pyramids whose every brick is a whole number from 1 to the range, counted exactly at build time. The generator draws from all of them with equal chance, which was checked by drawing until every one at the smallest settings had appeared. Shaded cells are the ones the builder will not use.
What the builder does when a size and a range do not fit
It keeps your range, because that is the arithmetic you chose, and uses a smaller pyramid. A base of 5 at up to 20 is possible but there are only 17 such pyramids, too few for a pack of sheets that should not repeat, so it counts as not fitting. Magic squares have the same problem from the other side: the smallest total is 15 for 3 × 3 and 34 for 4 × 4, so a square at up to 10 moves to 20, and a 4 × 4 at up to 20 becomes a 3 × 3. Every one of these changes is written under the preview, so a sheet never arrives different from the settings without saying why.
The Hard step is algebra, and every Hard sheet needs it
On a Hard sheet there comes a point where no sum has a single gap. The way through is to give one missing number a name. Call it n, write each neighbour in terms of n, and carry on until a sum that is already complete can be checked: that check is an equation, and it says what n is. The figures below are two real Hard sheets from this builder, stopped at the moment the letter resolves, with the working the solver wrote.
A Hard pyramid is an algebra lesson that never uses the word.
The letter has further to travel on bigger sheets. On a pyramid with a base of 6 at up to 1,000 it passes through 6.6 sums on average before it resolves, and on a 4 × 4 square through 4.4. The solver was first written with general elimination, which could combine seven or eight sums at once into a single step; that is a proof, not something anybody does with a pencil, so the Hard step is one letter carried through single gaps, and a sheet that would need two letters at once is never printed.
What every sheet is checked for before it prints
- One answer, proved. The solver only ever writes a number that follows from the ones already on the sheet, so reaching the end proves nothing else could go there. The engine was also checked against a separate rank calculation that does not use the solver at all.
- Every printed number is needed. The generator starts from a finished puzzle and removes numbers in a random order for as long as the level's own steps still finish it. Take any printed number away and the sheet stops working.
- The level is real. A Medium pyramid is checked to be impossible by adding alone, and a Hard sheet to be impossible without the letter step. When a sheet is not, one printed number is moved to another square and the check runs again, which is what rescues the three-wide Hard pyramid: it has exactly one pattern that needs the letter, both bottom corners and the top.
- No pack repeats a puzzle while different ones are left. At the smallest settings they can run out, and the builder says how many repeated rather than printing them quietly.
Which settings suit which sums
Year groups and grade levels do not line up between countries, so this is organised by the sum being practised. The right-hand column is measured, not estimated.
| Practising | Settings | What the sheet actually asks |
|---|---|---|
| Adding within 10 | Pyramids, base of 3, up to 10, Easy | Only 50 different pyramids exist at this setting, and 14% of their sums make exactly ten. |
| Number bonds to 20 and missing addends | Pyramids, base of 4, up to 20, Medium | Medium means some bricks have to be worked backwards, which is the missing-addend question in disguise. 19% of the sums cross a ten. |
| Adding three small numbers | Magic squares, 3 × 3, up to 20, Easy | Every line is three numbers to one total, 83% of lines cross a ten, and only 32 different squares fit under 20, so the variety comes from which squares are left blank. |
| Two-digit adding and subtracting with regrouping | Pyramids, base of 4, up to 100, Medium | 40% of the sums on these sheets carry, across six sums to a pyramid. |
| Three-digit column arithmetic | Pyramids, base of 5, up to 1,000, Medium | 63% of the sums carry, and a base of 5 means ten sums to a pyramid. |
| A first taste of algebra | Pyramids, base of 3, up to 20, Hard | Small numbers on purpose, so the only new idea is calling a missing brick n. At this size the letter resolves after 3 sums: both bottom corners and the top are printed, and the middle is what is left. |
How to solve a pyramid, if you never have
- Look for two bricks side by side that are both filled in. The brick resting on them is their total.
- Look for a filled brick with one filled brick under it. The other brick under it is the difference.
- Every number you write opens up a neighbour, so go back over the pyramid after each one.
- If no brick has a single gap, call one missing brick n and write the bricks around it as n plus or minus something.
- Keep going until a brick you already know has an expression next to it. Set them equal, and that gives n.
What this page does not do yet
Only whole numbers, only addition, and only these two puzzles. Multiplication pyramids, decimals and negative numbers are the obvious next options, and cross-number puzzles and target-number sheets were considered and left out of this first version: a target-number sheet usually has many right answers, and every sheet here is built around there being exactly one.
The magic squares do not use the classic rule that each number appears once from 1 to 9, because that allows almost no variety. Every square does hold different numbers. The 4 × 4 squares are built by scaling and shifting the 7,040 arrangements of 1 to 16, so their numbers are always evenly spaced, which a sharp-eyed solver can use.
How this page is made and checked
Print Puzzles is part of the Hamilton Digital Media Limited puzzle network. This page is written and reviewed by Brian Hamilton and Karan Hamilton. The engine was checked with a harness that generated puzzles of both kinds at every size, range and level, and confirmed for each one that every sum holds, every number is inside the range, the solver reaches the printed answer exactly, every printed number is needed, the level cannot be finished one step lower, and an independent rank calculation agrees the answer is unique. It also counted every 3 × 3 square and every small pyramid by brute force to check the figures in the tables above. Read more about how we make puzzle packs, or tell us about a sheet that misbehaves through the contact page.
Nothing here costs anything and there is no account. Puzzles are made in your own browser, and the terms of use let teachers, tutors, clubs and families print and copy these sheets and their answer keys for a whole class.