Those of you who have been subscribers from the start may remember a set of four puzzles that appeared in issues 5, 13, 20 and 28 called Quadratum. All of them were fully fledged sudokus and the first two appeared before The Times had even heard of them. The third appeared a couple of months after they published their first. When I submitted QIV I said that it would be the last in the series as sudoku had now taken off.

Roll on to last year when The Magpie published Domino Thief by Zag, my co-editor of CQ. He resurrected the idea of the thieving magpie which I’d used in issue 11 of The Magpie. This got me thinking that if a thieving magpie could be resurrected after 200 odd issues, then so too could Quadratum. As luck would have it Chalicea had submitted a few puzzles for CQ that we published which involved chess knights on a 6×6 grid. It was a nice idea and I decided to see if I could use knights touring a sudoku grid.

Of course, there are many knights’ tours on a chessboard – Arden has set a few for The Magpie. However, a sudoku is 9×9 and so it is impossible for a knight to make a tour on a board of this size – if the dimensions of a board are both odd then it’s impossible. However, I noticed that you can tour the perimeter cells in a 3×3 block and I wondered if that could be used.

I made many copies of a 9×9 sudoku grid looking at the various tours that could made round the perimeters of the 3×3 blocks. In puzzles that involve knights moving as in chess it is useful to make the grid into a checker board as on each move a knight moves to a square of the opposite colour. There are two types of tour possible in a 3×3 block. One tour starts in a corner cell and ends in a middle cell that’s not adjacent to the start cell, in other words the opposite sides. The other tour starts in a middle cell and ends in a corner cell that’s not adjacent to the start cell, in other words the opposite corners. This gives rise to two routes for both starting positions. In the following diagrams below, we start at 1 and count along to 8.

A corner start exhibits mirror symmetry on a diagonal whilst a middle start again has mirror symmetry but this time it is in a horizontal or vertical axis. For any 3×3 block, starting in a corner means that you end in a middle which in turn forces the next block visited to start in a middle cell and finish in a corner. Starting in a middle cell means that the finishing cell must be in a corner which in turn forces the next block to start in a corner and finish in a middle. So, we have a pattern CM, MC, CM, MC … or MC, CM, MC, CM ….

Armed with this information I set about finding a tour with the knight always completing a block before moving on to the next one. It seemed sensible to start in block A and to be the middle cell of the 3rd row which forced the tour to go in to block B. Of course, some parts were forced but others weren’t and gave rise to two or three possible routes. However, knowing the end cell in block E and working backwards in that block gives the end cell for block D and soon I had a tour. AJ has already covered this in his solution notes.

Now onto the clues. I decided that the knight would collect digits that when concatenated would form 2-digit numbers. It was then a case of looking at various sets of numbers, primes, squares, triangular etc that contained all the digits from 1 to 9 inclusive. This was really good fun and most enjoyable – honestly. I started with the primes in block A and I wrongly surmised that that would force a 5 in the middle cell. My test solver pointed this out by supplying a list of a further eight, half of which had a 2 in the cell and the other half an 8! Thankfully block E got me out of that predicament and I just had to reorder a bit of my logical solution pathway.

But that’s getting ahead of myself. I chose a prime set for block A and opted for them being in ascending order and similarly for the squares in block E. I like to fit something interesting into my sudoku puzzles and previous puzzles have had 9-digit squares and sums like 218 + 349 = 567 appearing. Now, there are 39 possible products involving three 3-digit numbers that contain all the digits from 1 to 9 inclusive and give a 9-digit answer that also has this property. I chose one and positioned it in the grid. Solvers with long memories may remember QIV – The Manticore which also had this as the endgame although with a different set used. I’d already decided that the lettered cells would contain different digits. So, with blocks A and E entered along with row 8 and the lettered cells in was a case of solving the resultant sudoku.

With that done it was just a case of looking at the 2-digit numbers that they generated and finding suitable clues for the blocks.

This was my first version of this type of puzzle – a knight moving around the perimeter cells of the 3×3 blocks in a sudoku – and it contains full information which is something that I like to do with a first puzzle in a series. ( I already knew that there would be many others as it is a rich vein to mine ). It has the advantage of giving solvers everything they need to try and solve the puzzle. The drawback of course is that for some solvers it’ll be too easy and not much of a challenge. However, I like to try and encourage people to give the numerical puzzle a go. There will always be other puzzles that will satisfy those that like their puzzles on the ‘well-done’ side. As alluded to above there are other versions of this style of puzzle, one of which appeared in CQ, and involved two knights with some entries/blocks unclued which was far more demanding.

Finally, if you haven’t heard Keith Emerson’s version of Living in the Past by Jethro Tull played on a Hammond and Moog then you should. It’s quite simply amazing. Pure genius!

Posted on Friday, November 10th, 2023 at 4:04 pm under Setting the scene.
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