When a Suguru puzzle runs out of singles
Most Suguru boards go down in one long run of forced cells. You read a cage, you see the one digit that fits, you write it, and the cell next door opens up. Then you meet a board that stops. Twenty cells are filled, sixteen are empty, and every cage you look at has two digits that both still fit. Nothing is wrong with the board. You have simply run out of the one move you have been using, and there is a second move waiting.
This is a worked example of that second move on a real 6×6 board, with the complete answer at the end. The puzzle is the one sold as Number Blocks, and known just as widely as Suguru, or as Tectonic — same rules, three names. If you want the rules from scratch rather than the technique, how Number Blocks puzzles work is the page for that.
The board
Two rules run the whole puzzle. A cage of N cells holds the digits 1 to N, once each. And two cells that touch — including diagonally — never hold the same digit, even when they sit in different cages.
The board shown is 6×6, carved into nine cages, with ten digits printed. Each cell shows its cage letter in small type; a large digit is one of the printed givens.
The cages, written out: A is r1c1, r1c2, r2c1 (three cells, so it holds 1–3). B is r1c3, r2c2, r2c3, r2c4, r3c2 (five cells, 1–5). C is r1c4, r1c5, r1c6, r2c5, r2c6 (five). D is r3c1, r4c1, r4c2, r5c1, r5c2 (five). E is r3c3, r3c4, r4c3, r4c4, r4c5 (five). F is r3c5, r3c6, r4c6 (three). G is r6c1, r6c2 (two cells, so it holds 1 and 2). H is r5c3, r5c4, r6c3, r6c4, r6c5 (five). I is r5c5, r5c6, r6c6 (three).
What the singles get you
Start where the cages are small, because a small cage tells you its whole digit set before you place anything.
Cage F has three cells and already shows its 3 at r3c5, so r3c6 and r4c6 share the 1 and the 2. The cell r4c6 sits directly above r5c6, which holds a printed 2, so r4c6 takes the 1 and r3c6 takes the 2. Cage I works the same way: it needs 1, 2 and 3, its 2 is printed at r5c6, and r5c5 touches the printed 1 at r4c4 diagonally, so r5c5 is the 3 and r6c6 is the 1. Cage G is two cells holding 1 and 2, and r6c2 sits next to the printed 1 at r6c3, so r6c2 is the 2 and r6c1 is the 1.
Four more cells come from asking the question the other way round — instead of “what can this cell hold”, ask “where can this cage put this digit”. Cage D still needs a 4, and the printed 4 at r3c2 touches r3c1, r4c1 and r4c2, so the only cell left for it is r5c2. Cage H still needs a 3, and the 3 you just placed at r5c5 blocks r5c4, r6c4 and r6c5, leaving r5c3. That 3 at r5c3 then blocks cage E’s 3 out of r4c3, the printed 3 at r3c5 blocks it out of r3c4 and r4c5, and r3c3 is what remains. And the 3 at r3c3 pushes cage B’s 3 up to r1c3.
That is ten placements on top of ten givens, and here is where it stops. Large digits are settled; small pencil marks in each open cell are everything it can still legally hold.
Look down that board for a cell with one digit left, and there is none. Look through the nine cages for a digit with one cell left, and there is none of those either. Both versions of the single are used up with sixteen cells still empty.
The move that restarts it
Take cage A on its own. It is three cells, so it holds 1, 2 and 3. Its 3 is printed at r1c1, which leaves the 1 and the 2 to share r1c2 and r2c1 — and nothing on the board yet says which way round.
Now look at r2c2. It sits directly below r1c2 and directly right of r2c1, so it touches both of the cells that are going to hold cage A’s 2. Whichever of the two takes that 2, r2c2 is sitting next to it. So r2c2 cannot be a 2, and you know that without ever settling which cell the 2 goes in.
That is the whole technique: when every cell that could hold a digit in one cage touches the same outside cell, that outside cell loses the digit. It is a deduction about a cage rather than about a cell, which is why scanning cell by cell never turns it up.
The payoff is immediate. The cell r2c2 was down to a 2 or a 5, so it is the 5. Cage B then has only the 2 left for r2c3. The 2 at r2c3 touches r1c2 diagonally, so r1c2 takes cage A’s 1, r2c1 takes the 2, and from there the last twelve cells fall over in straight naked singles, one digit forced after another until the grid is full.
Two other eliminations of exactly this shape are sitting on the same position, in case you want to find one yourself before reading on: one on cage B’s 2, and one on cage E’s 5.
The answer
Read it against the cage list and both rules hold everywhere: each cage carries 1 up to its own size, and no two touching cells repeat a digit in any of the eight directions.
How often a board asks for it
The board above came out of Puzzle Maker Pro at 6×6 on the Hard setting. To see how typical it is, here are 360 boards from the same engine — sixty at each of six settings — checked cell by cell against both rules, counted for solutions by exhaustive search, and then solved by a technique tracer that records how far up the ladder it had to reach.
All 360 hold both rules and have exactly one solution. What separates the settings is how many digits get printed, and whether the cage elimination is on the menu at all.
| Setting | Printed givens (median, range) | Boards that needed a cage elimination |
|---|---|---|
| 6×6 Easy | 12 (12–14) | 0 of 60 |
| 6×6 Medium | 9 (5–14) | 0 of 60 |
| 6×6 Hard | 8 (4–14) | 42 of 60 |
| 8×8 Easy | 21 (21) | 0 of 60 |
| 8×8 Medium | 14 (10–19) | 0 of 60 |
| 8×8 Hard | 12 (8–18) | 55 of 60 |
Easy and Medium are the same solving job at two densities: singles all the way down, with Easy printing enough digits that the next one is usually in sight and Medium making you hunt for it. Hard is a different job. On the larger grid it asks for the cage elimination on nine boards in ten, and it prints fewer digits than Medium does while doing it — which is worth knowing if you assumed the given count was the difficulty.
If you are printing these
Three things follow for a book or a worksheet pack.
Band a volume by grid size as well as by setting. The sizes run 5×5, 6×6, 7×7, 8×8, 9×9, 10×10 and 11×11, plus 5×10 and 10×5 for a tall page. Grid size is the difference a shopper can see in a thumbnail; Easy against Hard is a difference they find out about on page one.
Put the technique somewhere. A buyer who has only ever met singles will stall on a Hard page exactly where this article stalled, and a short solving note at the front of the book turns that stall into the reason they bought it. The section above is the whole explanation — one cage, one outside cell, one digit removed.
Use all three names. The Titles tab offers Number Blocks, Suguru and Tectonic, each on its own or with the grid size and the difficulty appended, so one run can carry the name your shelf searches under. Every board arrives with its completed grid as the answer key, which is the page that has to be right in a book where every puzzle claims a single solution.
Every paid Puzzle Maker Pro edition licenses you to sell the finished books, worksheets and packs you make to your own readers and customers. Productivity adds the right to sell or supply the puzzles you generate to other publishers, businesses and sellers.
Where this sits on the shelf: the print side is covered in number puzzle books for KDP and the pack side in number puzzle printables for Etsy. The settings that produced the board above are in how to create Number Blocks puzzles.