# Prize crossnumber, Issue 18

Can you solve it?

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

### 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.

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