It's assumed that numbers don't have leading zeroes. 1 across, 2 across: 7997 and 7887 each have ony one prime facor in the range 100 to 999. 4 across: first digit is 1,2,3 or 4, remaining digits add up to 4- first digit; only 2 values like this are multiples of 7: 112 and 301. 4 down: palindromic multiple of 3, only two digits - only 33,66, and 99 are possible; so first digit of 4 acoss is not 1, so 4 across is 301 and 4 down is 33. 9 across: mean of two consecutive fib nos is half the following fib no, so this is half a fib no. The only even fib nos in range 20 to 200 are 34 and 144, so this is 17 or 72. 10 across: half of a cube, multiple of 27; 5 down tells us it's the interior angle (degrees) of a regular polygon so cube is in in range 200 to 358 so less than 27*27 so must 8*27 which is 216, so the angle is 108 degrees. 5 down: 9 times the number of sides of regular polygon whose interior angle is 108 degrees is 45, so the interior angle of this polygon is 180-360/45 = 172 degrees 9 across: ends in a 2, so it isn't 17 it's 72. 15 across: two digits, 3 expressions as sum of a triangular number and a square, so 5 possibilities: 19, 46, 64, 82, 91. 18 across: product of 2 primes or cube of a prime, range 100 to 999. 19 across: a cube in range 100 to 999. 20 down: 2 digit multiple of 4 whose digits add up to 15. two figits to 15 are 7 and 8 or 6 and 9, of the four numbers that can be made only 96 is a multiple of 4 - so 96 it is. 8 across: second digit is 9, and its less than 20% of 20 down which is 96, so it's 19 or less - so 19. 21 down: LCM of two primes is their product, so 23*29 which is 667 21 across: longest site of a right triangle, 1st digit is 6 (from 21 across), all sides have two digit lengths, looking at the tree of pythagorean primitives we have 3 primitive candidates and can get 4 more as multiples of primitives - the following 7 triples. Elimination of the misfits is quite easy (see 12 acr, 6 acr below) A 36 48 60 B 11 60 61 C 16 63 65 D 25 60 65 E 33 56 65 F 39 52 65 G 32 60 68 12 across is half of 8 down, so first digit is in range 5 to 9. according to 7 down it is length of the middle side of that triangle, which eliminates A 22 across: 4 digit prime ,digits in descending order, second digit 6: possibles are 7621, 9631, 8641, 7643, 9643. 6 across: range 1000 to 9999, power of 17, last digit is 3: so it's 4913. That means the shortest triangle side starts with 1, so its triangle B or triangle C. Triangle B violates the no leading zeroes assumption. So it's triangle C. 7 down is 16, 12 across is 63, and 21 across is 65; so 8 down is 126. 8 down is the mean of 2 down, 4 down, and 15 across, so 378 = 239+33+18acr, so 18acr = 106 (which has 4 factors, 1,2,53,and 106, a useful check). 17 across: as the big angle is 106, isoceles trangle the other angles are each 37 degrees. 14 down: power of 71 in range 1000 to 9999 is 5041 9 down is square of two digit 11 across, begins with 7 so is either 729 or 784. 2nd digit of 9 down is seond digit of 11 acoss, ans 7 isn't 2 so we have 28 and 784. 15 across begins with 4 (from 9 down) which eliminates the possibles except 46. 16 down is 16 less than a ,ultiple of 61, and is three digits 6x5.so the multiple of 61 ends with 1 and is bewteen 621 and 711, so that's 671. 16 less is 655. 3 down: half of 15 across plus 2/7 of 8 down is 23+36 = 59. 25 across: the only 3 digit palindrome which has 7 as the second digit (from 21 down) and is a multiple of 11 is 979. 6 down: there appears to be nothing constraining this except the two values it crosses, that it's a multiple of 11, and that 24 across is a factor of it. 22 down is (9across)%, ie 72%, of 19 down. It's first digit is > 6 because 22 across has digits in descending order and has 2nd digit 6 (from 20 down). If 72% (18/25) is exact, 19 down must be a multiple of 25 but it's a (3 digit) cube less than 100/0.72 so 19 down is 125 and 22 down is 90. 22 across is 96_1 where the missing digit is a factor of is a factor of 18 and is in range 2 to 5, so it's 9631 (and 9631 is a prime, as required). 23 across is 9632/301 = 32 13 down: the across clues have provided all the digits (338), and it is more than 110% of 4 across 6 down is divisible by 24 across, which is divisible by 10, so 6 down is 4x00 (1st and 3rd digits known from across answers), so it's divisible by 100; it's defined to be divisible of 11, so it's divisible by 1100. the only 4 digit number beginning with 4 and divisible by 1100 is 4400, so 6 down is 4400. 24 across is 5_0 and a factor of 4400 so it's 550. So the clues determine a unique solution.