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Answers to puzzles on square grids

These are the answers to the puzzles which appeared in this blog post.

3×3 Grids

Puzzle #1

4 5 8 even
6 1 3 prime
7 2 9 cube
prime cube prime

Puzzle #2

3 8 9 emirp
4 2 6 even
7 5 1 emirp
emirp odd square

Puzzle #3

3 9 1 multiple of 17
7 2 5 multiple of 25
4 8 6 multiple of 9
multiple of 11 multiple of 16 multiple of 12

Puzzle #4

4 6 3 prime
8 2 9 prime
7 5 1 prime
prime square prime

2×2 Grids

Puzzle #5

4 1 prime
3 2 multiple of 8
prime multiple of 4

Puzzle #6

No two digit prime number can end in 5 (as it would be divisible by 5). Therefore the 5 must go here:

5 prime
prime
prime prime

However, there are only two prime numbers beginning with a 5: 53 and 59. 9 is not available, so only 53 is allowed. But to complete the grid, two primes beginning with 5 must be formed.
Therefore it is impossible to complete the grid.

Puzzle #7

There are 22 ways to do this (or 11 if you ignore reflections).

And Finally…

Similar puzzles can be formed from non-square grids, like these:

Puzzle #8

There are two ways to do this:

1 7 prime
5 9 3 prime
prime prime prime
7 1 prime
5 9 3 prime
prime prime prime

Puzzle #9

As every two or more digit prime number is odd, the squares marked with a * must be odd:

* prime
* * * prime
prime prime prime prime

1, 3, 5 and 7 are the only available odd digits, so one of these four square must contain a 5. This however means that one of the numbers formed ends in a 5 and so is not prime.
Therefore it is impossible.

These are the answers to the puzzles which appeared in this blog post.

3×3 Grids

Puzzle #1

4 5 8 even
6 1 3 prime
7 2 9 cube
prime cube prime

Puzzle #2

3 8 9 emirp
4 2 6 even
7 5 1 emirp
emirp odd square

Puzzle #3

3 9 1 multiple of 17
7 2 5 multiple of 25
4 8 6 multiple of 9
multiple of 11 multiple of 16 multiple of 12

Puzzle #4

4 6 3 prime
8 2 9 prime
7 5 1 prime
prime square prime

2×2 Grids

Puzzle #5

4 1 prime
3 2 multiple of 8
prime multiple of 4

Puzzle #6

No two digit prime number can end in 5 (as it would be divisible by 5). Therefore the 5 must go here:

5 prime
prime
prime prime

However, there are only two prime numbers beginning with a 5: 53 and 59. 9 is not available, so only 53 is allowed. But to complete the grid, two primes beginning with 5 must be formed.
Therefore it is impossible to complete the grid.

Puzzle #7

There are 22 ways to do this (or 11 if you ignore reflections).

And Finally…

Similar puzzles can be formed from non-square grids, like these:

Puzzle #8

There are two ways to do this:

1 7 prime
5 9 3 prime
prime prime prime
7 1 prime
5 9 3 prime
prime prime prime

Puzzle #9

As every two or more digit prime number is odd, the squares marked with a * must be odd:

* prime
* * * prime
prime prime prime prime

1, 3, 5 and 7 are the only available odd digits, so one of these four square must contain a 5. This however means that one of the numbers formed ends in a 5 and so is not prime.
Therefore it is impossible.

Matthew Scroggs is a PhD student at UCL working on finite and boundary element methods. His website, mscroggs.co.uk, is full of maths and now features a video of him completing a level of Pac-Man optimally.
Twitter  @mscroggs    Website  mscroggs.co.uk    + More articles by Matthew

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