Puzzles

3 down is a multiple of 101, so its middle digit must be 0:

The clue 4 across now tells us that the units digit of today's number (ie 1 across) is 5 or 0. As 3 down cannot begin with 0, it must be 5.

3 down is a multiple of 101:

The sum of the digits of today's number is at most 9+9+5 = 23, so 1 down starts with a 1 or 2. This means that the sum of the digits of today's number is at most 2 + 9 + 5 = 16, so 1 down must start with a 1:

Two times today's number (ie 4 across) must start with a 2 or 3. If it started with 2, then the sum of the digits of today's number (1 down) would be 12, so today's number would be 165. The product of the digits of today's number (5 across) would be 30. This is not possible as 5 across ends in 5. The first digit of 4 across must therefore be a 3:

The sum of the digits of today's number is 13, and we can fill in the rest using the clues for 4 and 5 across:

Therefore, today's number is 175.

As 3D is 3 times 5A, and they both share a final digit, their final digit must be 5 or 0.

If it were 0, then 4A would also end zero, so 3D ends in 00. This would mean that 5A also ends 00. 2D therefore would end in 0, but then it's impossible for the sum of 2D's digits to be 19. Hence, the final digit of 5A cannot be 0, so must be 5:

As 4A is 2 times 5A, it ends in a 0:

3D is a multiple of 3, so must be 105, 405, or 705. These would mean 5A is 35, 135, or 235 (respectively). 5A has three digits so this rules out the first option. Both other options end in 35:

4A is 2 times 5A, so must end in 70:

The sum of the digits of 2D is 19:

If 5A were 135, then 4A would be 270. But then 1D ends with 21 and its digits cannot add to 15. Therefore 5A 235:

4A is 2 times 5A and 3D is 3 times 5A:

The sum of the digits of 1D is 15:

Therefore, today's number is 997.

2D is a multiple of 9, and so its digits add up to a multiple of 9. This means that 4A is a multiple of 9, and its digits add up to a multiple of 9 too. The digits of 4A can add up to at most 27 (if it's 999) and so must add add up to 9, 18, or 27. These would make 2D equal to 81, 162, or 243 respectively. 2D cannot be 81 (as it's a three-digit number); it also can't be 243, as this would mean 4A was 999, leading to a clash in the centre square. Therefore the sum of the digits of 4A is 18 and 2D is 162:

The sum of the digits of 4A is 18, so 4A is 369, 468, 567, 666, 765, 864, or 963. Of these, only 765 is a multiple of 5, and so 4A must be 765:

3D is greater than 2D, so the first digit of 3D is at least 2. If the first digit of 3D was 3 or higher, then 5A would be greater than 700, meaning 1D would be greater than 1000 and not a three-digit number. Therefore:

The last digits of 3D and 5A are the same, and so the last digit of 1D (and the first digit of 5A) must be even. 5A is greater than 500, and so it's first digit must be 6 or 8; it cannot be 8 as this would make 1D greater than 100. And so:

This only leaves one option for the last two digits to make 1D's clue correct:

Today's number is 812.

1D is "a square number"; 2D is "two times 1A"; 3D is "today's number": today's number is 564.

Today's number is 222.

Today's number is 333.

1 Found your far dented horn mixed to make square. (3) — 144 is a square number that is an an anagram of "Found your far dented horn" (one hundred and Forty four).

3 Eno back in Bowie's evening prime. (2) — 17 is a prime that is Eno backwards (onE) followed by a number in Bowie's evening.

4 Prime legs. (2) — 11 is a prime number and called legs in bingo.

5 Palindrome ends cubone, starts ninetales, inside poison ekans. (3) — 191 is a palindrome that is a the end of cubone the start of ninetales and inside poison ekans.

1 Odd confused elven elves hounded deerhound antenna. (5) — 11111 is an odd number that is an anagram of "elven elves hounded deerhound antenna" (eleven thousand one hundred and eleven).

2 Prime try of confused Sven with Beckham's second. (2) — 47 is a prime that is an anagram of "try of" (forty) and Sven with Beckham's second (Seven).

4 Prime even and Ian fed the being. (2) — 19 is a prime number that is the even letters of "and Ian fed the being" (nIneteen).