Make sure not to lose sight of the first clue, that every number is adjacent to one or both of its consecutive numbers.

Prime numbers between 1-25 are 2, 3, 5, 7, 11, 13, 17, 19, 23.

**(Back to puzzle)**

**(1/25) **

1 tells you that E2 is 22 and gives you what is henceforth called “the main rule” of the puzzle.

**(2/25) **

1 must be adjacent to 2, according to the main rule. 22 tells you that no number in Row 2 is prime, so the 02 must be in B1.

**(3/25) **

2 tells you that E5 is 08.

**(4/25) **

Since there are five multiples of 5 (5, 10, 15, 20, 25), 8 tells you that every number in Column C is one of them. Since C3 has to be prime (22) and only one multiple of 5 is prime, C3 must be 05.

**(5/25) **

5 tells you that 11 is B4, C4, or E4. E4 does not work, according to 22, because 11 is prime. C4 does not work, according to E5. So B4 is 11.

**(6/25) **

E1 cannot be a prime number, according to 22, so according to B4, A5 is 17 and E1 is 04.

**(8/25) **

Given where 5 is, 4 must be adjacent to 3, according to the main rule. D2 is not prime, according to 22, so D1 must be 03.

**(9/25) **

C1 and C2 are both multiples of 5, according to 8, so according to D1, D2 is 16.

Since 16 is adjacent to 3, 17 must be adjacent to 18, according to the main rule. A4 is less than 17, according to 17, so B5 must be 18.

**(11/25) **

B2 is not a prime number, according to 22, so B5 tells you that B2 is 06 and B3 is 19.

**(13/25) **

18 is not adjacent to 19, so 19 must be adjacent to 20, according to the main rule. So C5 cannot be 20. According to 8, C5 is a multiple of 25 and according to B3, it must be greater than 18, so C5 has to be 25.

**(14/25) **

The only remaining single digit numbers are 7 and 9. So according to 25 and 22, E4 is 09 and D5 is 07.

**(16/25) **

You know from 8 that 10 is in Column C. Since it cannot be adjacent to 9, it must be adjacent to 11. So 10 goes in C4.

**(17/25) **

19 is not adjacent to 18, so it must be adjacent to 20, which also has to be in Column C, according to 8. So C2 is 20.

25 must be adjacent to 24. So D4 is 24.

**(19/25) **

According to 3 or 8, C1 must be 15.

**(20/25) **

You can do this step earlier, but it’s easier to see now that the only remaining numbers are 12, 13, 14, 21, and 23. According to 17, then, A2, A3, and A4 are 12, 13, and 14 respectively.

**(23/25) **

21 and 23 are left. 22 tells you that 21 must go in E3, with 23 in D3.

**(Back to puzzle)**

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