How to solve Hidoku and Numbrix
Hidoku, also called Hidato, was invented by the Israeli mathematician Gyora Benedek. Numbrix is the version without diagonal moves, created by Marilyn vos Savant for Parade magazine. Both ask you to link the numbers in one unbroken path.
Start here
The rules
Fill the empty cells so that the numbers run in order from the first to the last, each number in a cell touching the one before it and the one after it. Every cell is used once.
In Hidoku, consecutive numbers can touch across, up and down, or diagonally. In Numbrix they can only touch across or up and down, never diagonally.
Some numbers are already placed, and each puzzle has only one solution.
Hidoku and Numbrix strategies
Hidoku and Numbrix are the same idea with one difference. In both you fill the grid so the numbers run in one unbroken path from first to last, each number touching the one before it and the one after it. In Hidoku the path can also move diagonally. In Numbrix it can only move across or up and down. These seven techniques work for both, and each diagram says which puzzle it shows.
A few terms first. A cell touches another if the path could step between them: any of the eight cells around it in Hidoku, or the four beside it in Numbrix. A gap is a run of missing numbers between two numbers that are already placed. A step is one move along the path, so 7 to 11 is four steps.
The diagrams use a small 6×6 grid so each step is easy to see. Our books use bigger grids, from extra-large Numbrix to Hidoku in fun shapes, but the reasoning is the same. Placed numbers are black, green marks the numbers the step gives you, blue marks cells still in the running, red with a cross is a cell a number can't go in, and the thin grey line shows the path where it's known.
You don't have to solve in order from 1 upwards. Fill whichever gap is easiest, anywhere on the grid, and come back to the hard ones when the cells around them have filled in.
Starting points
The number between two
Seeing it. Look for two placed numbers with one number missing between them, such as 3 and 5. The missing number has to touch both.
Using it. List the empty cells that touch both numbers. If there's only one, the missing number goes there. In the diagram, the cells touching both 3 and 5 are the one holding 9 and the green one, so 4 is green. In Numbrix there are never more than two cells that touch both, so this solves more often there.
Starting points
Counting steps
Seeing it. A number has a few cells it could go in. Check how far each one is from the next placed number along the path.
Using it. Count the moves from each possible cell to that number, the shortest way the puzzle allows. If it's more than the steps left, the cell is out. In the diagram 21 must touch 20, and there are three cells it could use. 21 to 24 is three steps, but from the crossed cell 24 is four moves away even with diagonal moves, so 21 can't go there. In Numbrix, count only moves across and up or down.
Starting points
A gap with one route
Seeing it. Two placed numbers are exactly as far apart on the grid as their gap allows, with no spare steps to go round anything.
Using it. The path has to take the shortest way, and if there's only one shortest way, fill it in. In the diagram 7 and 11 are four steps apart along a row, and the gap from 7 to 11 is four steps too, so 8, 9 and 10 go straight along. In Hidoku look for the same thing along rows, columns and diagonals, where a straight line is the only shortest route.
Dead ends
Dead-end cells
Seeing it. Look for empty cells with only one empty neighbour, usually in corners, along edges or next to blocks of placed numbers. Every number except the first and last needs two neighbours on the path: one lower and one higher.
Using it. A dead-end cell must hold a number next in line to one of the numbers it touches, with the other neighbour on the path going into its one empty neighbour. In the diagram the corner touches 15 and one empty cell, so it's 14 or 16. 16 is already placed, so the corner is 14, and 13 must go in the empty cell below it.
Dead ends
Never trap a cell
Seeing it. Before you place a number, look at the empty cells next to it. Every empty cell will need a way in and a way out.
Using it. If a placement leaves an empty cell that no number could fill, the placement is wrong. In the diagram 21 could go above 20 or to its right. Above, it would leave the shaded corner touching only 21 and 40, and no number is next in line to both. So 21 goes to the right.
Routes
Keep routes open
Seeing it. A number has two possible cells, and one of them is the only cell another gap can use.
Using it. Leave that cell for the gap that needs it, and put the number in its other choice. In the diagram 34 could go in either blue or green, since both touch 33 and 35. But blue is the only empty cell touching both 41 and 43, so it must hold 42. 34 goes in the green cell.
Routes
Work from the other end
Seeing it. You're stuck going up from a number because it has several cells to choose from. Try the gap from its other end instead, counting down from the higher number.
Using it. Placed numbers near edges and corners often have only one way to go. In the diagram 21 could step four ways. From the other end, 25 already touches 26 and 8, so 24 must be the cell to its left. The only empty cell touching 24 is the one to its left, so that's 23. The only cell touching both 23 and 21 holds 22. If neither end helps, move to another gap and come back: each number you place elsewhere takes cells away from the hard one.
