What Is a Number Puzzle? Definition, Kinds and Worked Examples

A number puzzle is a puzzle whose clues, rules and answer are all about numbers: how they count, add, multiply, line up in a row or balance on a scale. You solve it by reasoning about how numbers behave rather than by following a method you already know, and a well-made one has exactly one answer.

The arithmetic inside a number puzzle is often easy. The hard part is seeing which sums to do and why.

This article explains what counts as a number puzzle, its parts and main kinds, how it differs from a maths exercise and what makes a good one. Along the way it works through four small examples that you can check for yourself.

What are the parts of a number puzzle?

Like any puzzle, a number puzzle has a starting point, rules and a goal. What makes it a number puzzle is that all three are written in numbers.

• The unknowns. One or more hidden numbers: a missing term, a secret code, a set of tokens, or the one fake coin among real ones.
• The clues. Facts that tie the unknowns down, such as a sum, a product, a rule about digits or the result of a weighing.
• The range. The set of numbers that are allowed. Saying that the answer has two digits, or is a whole number, is part of the puzzle, not a detail.
• The question. One number, or one small set of numbers, to find.

The range does more work than it seems to. In the digit puzzle further down, the words two-digit number cut the field to ninety candidates before any clue is read, and the two clues then cut those ninety to one. Without a stated range, a solver cannot know when the search is complete.

What kinds of number puzzles are there?

Number puzzles come in many shapes, but most fall into a handful of groups. The table compares them by what the clues talk about and what you are asked to find.

The groups overlap. A fewest-pieces puzzle is part counting and part search. Some number puzzles even hide their numbers behind statements about what different people know, so they sit halfway between number puzzles and logic puzzles.

What does a sequence puzzle look like?

Here is a small one. By the simplest rule you can find, what comes after one, three, six, ten, fifteen?

Write the gaps between neighbouring terms underneath: two, three, four, five. Each gap is one more than the last, so the next gap is six and the next term is twenty-one. There is a second way to see the same rule: each term is the total of the whole numbers from one up to its place in the row, and the total from one to six is twenty-one.

Strictly speaking, no list of terms forces a single next term. Take the rule above and add the product of the place number minus one, minus two, minus three, minus four and minus five. That extra part is zero for each of the first five places, so the row you were shown does not change, but the sixth term becomes one hundred and forty-one.

So a fair sequence puzzle rests on an unspoken agreement that the simplest rule is the intended one. A good setter shows enough terms that the simple rule is clear and any rival rule looks forced. The riddle below is a sequence of exactly this kind.

How do digit puzzles work?

A digit puzzle describes a secret number through its digits. Try this one. I am a two-digit number. My digits add up to ten. Swap my digits and the new number is thirty-six more than me. What am I?

Swapping makes the number bigger, so the units digit is the larger one. Each step of difference between the two digits changes the swapped number by nine, and thirty-six is four nines, so the digits differ by four. Two digits that add to ten and differ by four are three and seven. The answer is thirty-seven, and seventy-three minus thirty-seven is indeed thirty-six.

Notice that each clue does real work. With only the sum clue, nine numbers fit, from nineteen up to ninety-one. With only the swap clue, five numbers fit: fifteen, twenty-six, thirty-seven, forty-eight and fifty-nine. Together the two clues leave exactly one. That is the mark of a tidy digit puzzle: no spare clues, and no spare answers.

What are missing-number and equation puzzles?

A missing-number puzzle hides one value inside a structure. Picture a pyramid of bricks with three bricks on the bottom row, two in the middle and one on top. The value on any brick equals the total of the pair of bricks it rests on. The bottom row reads four, blank, three, and the top brick reads twenty-five.

The middle bricks are four plus the blank and the blank plus three. Added together they make seven plus twice the blank, and that must equal twenty-five. So twice the blank is eighteen, the blank is nine, and the middle row reads thirteen and twelve.

Equation puzzles work the same way with pictures in place of letters. Each picture stands for a number, and you start with the line that holds only one kind of picture, then feed that value into the next line. Rod puzzles add a twist: the equation is written with sticks, and you move one stick to make a false sum true, so the clue lies in the shapes of the digits as well as their values.

What does a fewest-pieces puzzle look like?

Here is one to try. You have as many tokens as you like worth one, four and six. What is the smallest number of tokens that makes exactly eight?

The tempting move is to grab the biggest token first. Six leaves two, which needs two tokens worth one, so that route uses three tokens. But two tokens worth four also make eight, and that uses only two. No single token is worth eight, so two is the best possible, and four plus four is the only pair that works.

The point of the puzzle is that the obvious rule, biggest first, is not always right. To be sure of the answer you have to look at every way of reaching the total, or at least argue that no smaller set can exist. With the same tokens, the biggest-first rule also fails for nine: it takes a six and three ones, while four, four and one makes nine with only three tokens.

How can counting rule an answer in or out?

Some number puzzles are settled by a count before any search begins. The squares of side 1 up to 24 have a total area of exactly 70 × 70, yet computer work has shown that they cannot all be fitted into the square of side 70 ^[1].

Counting can also set a hard ceiling. A balance gives one of three results each time, so w weighings produce at most 3 to the power w different result strings, and no strategy can tell apart more coins than that ^[2]. With two weighings that is 9 strings, so two weighings can never pick out one odd coin from among ten or more ^[2].

The point for solvers is simple. A count can prove that a puzzle is impossible, or that a target is out of reach, but a count that comes out right only tells you the puzzle is worth trying.

How is a number puzzle different from a maths exercise?

The two can use the same arithmetic, but they ask for different things.

An exercise might ask you to multiply two numbers or solve two equations in two unknowns. The difficulty is in carrying out the steps without slips. A number puzzle often uses easier sums but does not tell you which sums to do.

The token example above shows the gap clearly. Every sum in it is small, yet the natural method gives the wrong answer, and the real work is noticing that a better set exists.

What makes a good number puzzle?

Good number puzzles share a few habits.

• One answer. The clues must rule out every other number in the range, so the solver knows when the work is done.
• A stated range. If the answer must be a whole number, a two-digit number or a number below some limit, the puzzle says so.
• No idle clues. Each clue should cut the list of candidates, as in the digit example above.
• The simplest rule is the intended one. In sequences, the intended rule should be clearly simpler than any rival.
• An idea, not a grind. The hard part should be a moment of insight, not a page of long sums.

When the range is finite, as in a digit puzzle or a fewest-pieces puzzle, you can check a puzzle completely by listing every candidate and crossing out those that break a clue. Setters can use the same check to make sure their puzzle has one answer and that no clue is wasted.

Which number puzzles can you try now?

Pick one of the puzzles below. Before you start, name its unknowns, its clues and its range, and when you finish, make sure no second number could also satisfy every clue.