From Clue A1, we know the diagonal must be made up of numbers 1,2,3,4 and 6. Since all corners are odd, cell E5 must be 1.
From clue A1, we know D4 must equal 2, 4 or 6. Now from clue E5, D4 can’t be 2 or 4, hence must be 6. Therefore, also from clue E5, B2 must be 4 and C3 must be
From clue C3, we know there are 3 even numbers on the 5th row, but since A5 is odd, the even numbers must be in B5, C5 and D5. From clue E5, this means D5 must be 8.
From clue E5 and given that D4 is 6, we know D1 must be 2 or 3, D2 must be 3 or 4 and D3 must be 4 or 5. Since A1 is 3, D1 must be 2, and since B2 is 4, D2 must be 3
From Clue D1, we know D2 is the only odd number in column D, so cell D3 must be 4.
From Clue D2 and A1, we know that A5 and E1 must equal 9 and 5. But from clue E5, we know A5 can’t be 5 so A5 must be 9 and E1 must be 5
From clue E1, we know 2 must appear in 4 columns. It is already in columns C and D. In column A, it can only be in A4 and in column E, it can only appear in E4. Since 2 can not be in both cell A4 and E4, we know it must therefore appear in column B. It can’t appear in rows 1 and 3, and we now know it will be in row for of column A or E, so 2 must be in cell B5
From clue B5, we see that C5 must be equal to 6
Previously, we found that 2 must appear in A4 or E4. But from clue C5, we know that A4 must be bigger than two, so we now know E4 must be 2
From clue E4, A3 and C1 are the same, and therefore can only be 1, 7, or 8. But from C5, A3 can’t be 1 or 8, so they must both be 7
From clue C5, we know that cell A4 must be 8
From Clue C1, we know there is only one odd number in column B. But cell B4 can’t be 2,4,6 or 8, so B4 must be the odd numbered cell and B1 and B3 must either 6 or 8. From cell D3, we now know B3 must be 8 and B1 must be 6
From clue B1, the remaining 6s must appear in A2 and E3.
C2 can only be 1,5,8 or 9. But given the constraint given in A2, we know C2 must be 1, and C4 must be 3
From Clue E3, we know E2 must equal 9
And finally given clue B2, we know that B4 must be equal to 7
