Our original prize crossnumber is featured on pages 52 and 53 of Issue 18.

- Download Crossnumber #18 as a PDF, or read on!
- Submit your answer

### Rules

- Solvers may wish to use the OEIS, Wikipedia, Python, send a telegram to your old maths teacher, etc to (for example) obtain a list of cube numbers, but no programming should be necessary to solve the puzzle. As usual, no numbers begin with 0.
- One randomly selected correct answer will win a
**£100 Maths Gear goody bag**, including non-transitive dice, a*Festival of the Spoken Nerd*DVD, and much, much more. Three randomly selected runners up will win a Chalkdust T-shirt. Maths Gear is a website that sells nerdy things worldwide, with free UK shipping. - To enter, submit the
**mean of all the digits in the row marked by arrows**using this form by**14 April 2024**. Only one entry per person will be accepted. Winners will be notified by email and announced on our blog by 1 June 2024.

### Crossnumber

Crossnumber #18, set by Humbug:

### Across clues

- 1. The mean of 9A and the square of 9A. (3)
- 3. Each digit* of this number is the mean of the digits before and after it. (10)
- 8. The mean of 12D and 21. (3)
- 9. The mean of 10A and 42A. (2)
- 10. The mean of 9A and 41A. (2)
- 11. The mean of 8 times 1A, 900 times 1A, and 1A. (5)
- 13. The geometric mean** of this number’s digits is 6. (6)
- 14. Each digit* of this number is the mean of the first and last digits of this number. (5)
- 16. The mean of 34A and 34. (2)
- 18. The mean of this number’s digits is 9. (2)
- 20. Each digit* of this number is the mean of the digits before and after it. (4)
- 22. The mean of this number’s digits is 7. (6)
- 25. Each digit* of this number is the mean of the digits before and after it. (4)
- 28. The mean of this number’s digits is 4. (4)
- 29. The mean of this number’s digits is 2. (6)
- 30. Each digit* of this number is the mean of the digits before and after it. (4)
- 32. A multiple of the geometric mean** of the first three digits of 15D. (2)
- 34. The mean of 16A and 16. (2)
- 36. The mean of 3 times 6D and 6D. (5)
- 38. Each digit* of this number is the mean of the first and last digits of this number. (6)
- 40. Each digit* of this number is the mean of the digits before and after it. (5)
- 41. The mean of 10A and 9A. (2)
- 42. The mean of 10A and 41A. (2)
- 43. The mean of two three-digit cubes. (3)
- 44. Each digit* of this number is the mean of the digits before and after it. (10)
- 45. The mean of 28A and 32A. (3)

### Down clues

- 1. Each digit* of this number is the mean of the digits before and after it. (10)
- 2. Each digit* of this number is the mean of the digits before and after it. (5)
- 3. Three more than the mean of the digits of 40A. (2)
- 4. Each digit* of this number is the mean of the first and last digits of this number. (4)
- 5. The geometric mean** of this number’s digits is 4. (6)
- 6. The mean of 6D and 6D. (5)
- 7. The geometric mean** of this number’s digits is 3. (3)
- 12. The mean of two times 12D and zero. (3)
- 13. The mean of two times a square and zero. (2)
- 15. The geometric mean** of this number’s digits is 2. (6)
- 17. The mean of this number’s digits is 6½. (6)
- 19. Each digit* of this number is the mean of the digits before and after it. (10)
- 20. Each digit* of this number is the mean of the digits before and after it. (4)
- 21. Each digit* of this number is the mean of the digits before and after it. (4)
- 23. The mean of 12D and 19. (2)
- 24. The mean of this number’s digits is 6. (2)
- 26. The mean of this number’s digits is 5. (2)
- 27. The mean of this number’s digits is 3. (2)
- 31. The mean of this number’s digits is a square. (6)
- 33. The geometric mean** of this number’s digits is equal to the product of the last three digits of 15D. (3)
- 35. The geometric mean** of this number’s digits is 0. (5)
- 36. The mean of eight times a prime and 0. (2)
- 37. The mean of 301 times 23D and 23D. (5)
- 39. Each digit* of this number is the mean of the digits before and after it. (4)
- 40. The mean of this number’s digits is 8. (3)
- 43. The mean of a two-digit prime and a two-digit cube. (2)

* Each digit excluding the first and last digits.

** The geometric mean of an $n$-digit number is the $n$th root of the product of the number’s digits.