Suguru is a captivating number puzzle. At first glance it looks like a broken Sudoku grid. But whilst it shares some commonalities with its more well-known cousin, Suguru is a different kettle of fish altogether. If you’re new to Suguru, have a look at our tutorial here to get started.

One of the techniques that’s fundamental to solving harder Suguru puzzles is *elimination by implication*. We are going to look at some examples.

To start us off, we’re using a puzzle taken from A Year of Suguru (Easy to Intermediate) - the puzzle is from June 14 if you have the book and want to play along, and complete the rest of the puzzle yourself.

Here’s the grid as presented in the book:

Let’s begin with an interesting example near the middle of the puzzle. I’ve highlighted two regions, green and red. The red region has no clue numbers in it at all, but we can still use it to solve the green cell with the **?** inside:

We can solve this cell purely through implication. We know that the red region must contain the numbers 1, 2 and 3 (because it contains three cells). It doesn’t matter what order they appear in that region, they will all affect the highlighted green cell because they are all adjacent to it either vertically or diagonally. That means our green cell (the one with the **?**) cannot contain a 1, a 2, or a 3. It also cannot contain a 4 because that already appears in the green region. Therefore the only number that can go there is a 5.

Even though the neighbouring region contained no clue numbers, its size and position made it useful. This was an example of simple implication. The *implied* contents of one region allowed us to directly solve a cell in a neighbouring one. But implication can go further.

Let’s turn our attention to the bottom right of the grid. I’ve highlighted four regions that have some interesting interplay.

We’ll begin with the green region. Given the presence of the 2 in the blue region above it, we know that the top cell of the green region cannot contain a number 2. Therefore the number 2 must go in one of the two bottom cells. This is a simple implication.

Because we now know that one of those two cells has to contain a 2, we can *further imply* that the left-most cell in the yellow region cannot be a 2. It doesn't matter which of the green cells is a 2, they both prevent the left-hand yellow cell from containing a 2. As there are only two cells in the yellow region, the left-hand cell must be a 1, which leaves the right-hand cell as a 2.

We’re not done yet. We can also infer, using the green region and the yellow one, combined with the blue one, that there’s only one place to put a 2 in the red region:

Here’s where it gets fun: the 2 we just filled in inside the red region now has a knock-on effect on the green region, telling us which of our two candidate cells must contain a 2 – the bottom one, it’s the only possibility.

Just by inferring that one of the bottom green cells must contain a 2, we’ve been able to fill in a neighbouring 2, which then allowed us to figure out another neighbouring 2, which in turn let us come back full circle and figure out which of the green cells must contain the 2! This is an implication chain at work, albeit a small one. They can get much, much longer.

Here’s another example of an implication chain. This is the May 25 puzzle from the same book as above, if you want to play along. Here’s the empty grid, and I’ve highlighted three green cells that we can fill in through implication initiated from the red region. Can you work out how?

We’ll start with the 3 in the red region. This is a handy *corner* cell which eliminates three neighbours in the region just to the right. We can imply the position of the 3 in that region, narrowing it down to two possible cells, like this:

Even though we don’t know which cell contains the 3, we *do* know that either way, we can eliminate two cells in the region below. Furthermore, the 3 in the neighbouring yellow region knocks out two more cells, leaving only one place to put our 3:

That 3 then gives us the position of the 3 in the region to the right:

That in turn tells us where the 3 goes in the next region up:

Phew - after all that, even though we didn’t yet work out the position of the 3 in the first region, we could still use its implied position to fill in three other cells in the chain.

Implication chains are a powerful tool for solving more difficult Suguru puzzles. Even if a simple implication doesn’t let you fill in a cell, it’s worth trying to follow the chain because you never know where it will lead. It may even lead back in a circle, letting you fill in the original cell.

Chains can often come in handy for getting you moving if you get stuck, but they are just one tool in the toolkit for Suguru-solvers. Don’t get hung up *only* looking at chains, when often simple elimination will get you to a result more quickly.