Conquer or be Conquered

Conqueror 1

It is recommended to view this puzzle as a document. You are also encouraged to print a physical copy.


Look at the hexagonal grid below which has two types of cells – clues (black) and answers (empty). Your job is to use the clues and follow these rules to fill in all of the answers.

  • Every answer must be a whole number between 1 and 50 (inclusive) and no number appears more than once. Since there are fewer than 50 answers, not every number from 1 to 50 will appear, and the smallest and largest number that appear may not be 1 and 50 respectively.
  • Each clue is in reference to all of the answers adjacent to it, i.e. its empty neighbours. Some clues may reference other clues. When referencing a clue cell in another clue you will see it in bold underline like A
      • For example if clue A is  “Sum is 36. All prime” this means that all of the answers adjacent to clue Asum to 36 and that they are all prime numbers.
      • Then if clue B is “Sum is 2 or 3 times A” This means that the answers adjacent to clue B sum to 2 or 3 times whatever the sum of the answers adjacent to A. In this case we know this is 36, so the answers adjacent to 2 add up to either 72 or 108.

WARNING: This puzzle is difficult! To complete it you will need to be good at maths and very persistent. You will need pen and paper to write out your workings. A calculator or a computer is not necessary, and you are discouraged from using either. Knowing about prime factorisation is essential, and knowledge of modular arithmetic is helpful.


  1. All primes, and all 1 away from a multiple of 6
  2. Sum to 7 more than J
  3. Only use the digits 1 and 3
  4. The smallest and largest answer in the grid and their average. The largest is a prime
  5. Sum to a multiple of 23. All but 1 are 1 away from a square
  6. Two numbers and their sum. One of the smaller numbers is a multiple of 10
  7. Sum is twice the sum of I. Sum is 11 times greater than the smallest at G
  8. Sum is 77. No primes
  9. Product is a square which ends in 600
  10. All divisible by 3 and sum ends in 1
  11. All squares
  12. Product ends in 0. All smaller than the smallest at G
  13. Sum is a power of 2.

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