Number patterns hiding inside picture puzzles
If the pictures make no sense, try counting them instead of looking at them.
A surprising share of puzzles that appear to be about shapes are really about numbers. The shapes are how the number is drawn, and as long as you look at them as pictures the rule stays invisible.
The tell is simple: you have run the visual checklist, found nothing, and the figures still look unrelated. That is the moment to start counting.
Things worth counting
- Elements — how many separate objects are in the cell
- Sides — a triangle is 3, a square 4, a pentagon 5. A sequence of triangle, square, pentagon is the sequence 3, 4, 5
- Corners — usually the same as sides, but not for curved or concave figures
- Line segments — including internal lines, which people skip
- Intersections — where lines cross
- Enclosed regions — closed areas inside the figure, the single most missed count of all
- Dots or marks — often decorative-looking and often the actual rule
Enclosed regions deserve special mention. A figure-eight has two. A triangle with a line through it has two. People see the outline and count one, and the rule vanishes.
Nothing about the shapes progresses in any visual sense — a triangle does not look like 'one less than' a square. Counted, it is the sequence 3, 4, 5, and the answer is a hexagon. This is the most common numeric disguise there is, and it is invisible until you count.
The differencing technique
When you have a sequence of numbers that looks irregular, write down the differences between consecutive terms and treat those as a sequence of their own. This single move solves a large fraction of otherwise impenetrable numeric puzzles.
- 2, 4, 6, 8 gives differences 2, 2, 2 — constant step
- 1, 2, 4, 7, 11 gives differences 1, 2, 3, 4 — the step itself progresses
- 1, 4, 9, 16 gives differences 3, 5, 7 — differencing again gives 2, 2, so it is quadratic. These are the square numbers
- 2, 6, 18, 54 has differences 4, 12, 36 which look no better — but the ratios are 3, 3, 3. When differences fail, try ratios
So the order is: differences first, ratios second. Between them they cover nearly everything you will meet.
Interleaved sequences
A sequence like 1, 10, 2, 20, 3, 30 is not one sequence behaving oddly. It is two sequences taking turns — 1, 2, 3 and 10, 20, 30.
The signal is alternation: terms that jump up and down rather than moving in one direction. When you see that, split the sequence into odd and even positions and look at each separately. A puzzle that was impossible becomes two puzzles that are trivial.
Sequences worth recognising on sight
- Squares: 1, 4, 9, 16, 25, 36, 49, 64
- Cubes: 1, 8, 27, 64, 125
- Primes: 2, 3, 5, 7, 11, 13, 17, 19
- Fibonacci: 1, 1, 2, 3, 5, 8, 13, 21 — each term is the sum of the previous two
- Triangular numbers: 1, 3, 6, 10, 15, 21 — the running total of 1, 2, 3, 4
- Powers of two: 1, 2, 4, 8, 16, 32, 64
Knowing these by sight is genuinely worth the ten minutes it takes. Recognising 1, 1, 2, 3, 5, 8 instantly costs nothing; deriving it under time pressure costs thirty seconds and often fails.
Fibonacci is the one to watch for because it does not respond to differencing — the differences of a Fibonacci sequence are another Fibonacci sequence, which looks like the technique failing. If differencing produces something that looks like the original, check whether each term is the sum of the two before it.
Combining numeric and visual rules
Harder puzzles run a numeric rule and a visual one at once — the count follows a progression while the shape cycles and the shading alternates. That is the stacked-rules situation, and the method is the same: give the count its own pass, in isolation, and write down what it does before you look at anything else.