It surprises me almost every time! I open up the Listener only to find a small grid and lots of letters and mathematical expressions in the clues. Ok, this time the clues had lots of numbers and brackets, but it was undoubtedly a mathematical puzzle… and alas, Ruslan’s last. Previously, this setter has entertained us with betting on the horses and Cleopatra’s needle!

A long preamble explained how all primes above 3 can be expressed as 6*n* + 1 or 6*n* – 1, the clues here giving the sum of two such values of *n* for two (not necessarily different) primes. The (hidden) answer is the product of the two values of *n*, with the entry being the lower *n* followed by the difference between the two. Thus 65, which is 5 × 13, is entered as 58. Answer and entry lengths are also given, the actual answer being irrelevant for solving the puzzle.

It seemed logical to start with the lowest sums for the two *n* values. 4dn and 19dn were both 2 followed by 10ac and 15ac, both 5, and 8ac and 16ac, both 6. A simple table was really all that was needed, once I had got my head around the basic concept. *u* and *v* are used for the two *n*s.

It was a just question of listing the two possible values for *u* and *v*, their associated primes (6*n* ± 1) and all the possible resulting entries. For example, if the clue were 5, then that could be expressed as 1 + 4 or 2 + 3. For 1 + 4, the primes are (5, 7) and 23 (not 25, obviously) and the possible entries are 518 or 716. For 2 + 3, the primes are (11, 13) and (17, 19) and the possible entries are 116, 118, 134 and 136. Some could be excluded as too short or long for the entry, others because they were not prime or didn’t fit with cells already completed in the grid. I entered all the possibilities for a given cell as I solved each clue, and then struck them out as crossing entries conflicted. I have included a sample from the table below.

At every stage of a mathematical, I expect to come across an anomaly that means I have to go through all my notes looking for an error. It is very satisfying as entries are slotted in the grid one by one. This was the case here… until I got near the end and found that 10ac/3dn could be either 134/674 or 136/676. Bugger!

Luckily, it didn’t take too long to realise that I had to check for “no two entries are the same”, and found that 136 was already in the grid at 15dn. It was then particularly satisfying to solve 12ac as 5839 × 6827 with the latter appearing in column 6.

An enjoyable last puzzle from Ruslan, and I’m sad there won’t be more.

Clue | u + v
| 6n ± 1
| Possible Entries | ||
---|---|---|---|---|---|

4dn 2 (2,2) length 2 |
2 = 1 + 1 | 5,7 5,7 |
50 52 70 |
||

10ac 5 (3,3) length 3 |
5 = 1 + 4 | 5,7 23, |
518 716 |
25 is not prime | |

5 = 2 + 3 | 11,13 17,19 |
116 118 134 136 |
|||

8ac 6 (3,3) length 3 |
6 = 1 + 5 | 5,7 29,31 |
524 526 722 724 |
||

6 = 2 + 4 | 11,13 23, |
1112 1310 |
too long | ||

6 = 3 + 3 | 17,19 17,19 |
170 172 190 |
|||

6ac 7 (3,3) length 3 |
7 = 1 + 6 | 5,7 |
532 730 |
||

7 = 2 + 5 | 11,13 29,31 |
1118 1120 1316 1318 |
too long | ||

7 = 3 + 4 | 17,19 23, |
176 194 |