Overview -------- - square pairs - x/y/k - opposite side - hypoteneuse - adjacent - column 4 - acute angle - speedrun Square Pairs ------------ The only square pairs are [11], [24], and [39], so r3c8-r4c9 is the 1-1. Pencil the other two X/Y/K ----- We immediately have x,y,k != 1 from column 9. x,y=2,3 gives 5k, 12k, 13k, which could work for some k<8 that's not even (5k fails), 5 (12k fails), 1, or 3. (So 7) Everything else is trivially dismissed: x,y=2,4 gives 12k, 16k, 20k, and you can't have a 0 x,y=3,4 gives 9k, 24k, 25k, and k can't take any value x,y=?,5 again requires a 0 x,y=2,6 gives ... 40k, again a 0 x,y=3,6 gives 27k, 36k, 45k, and k can't take any value ANything larger would require forbidden k=1 so x=2, y=3, and k=7 => The labels are 35, 91, and 84 => r5c4 is the [24] square pair => r1c2 is 3 and r2c3 is 9 Opposite side ------------- r5c9 is in [568], and r3c7 is max 9, so r4c8 is in [234], but not 4 because of GW, and r3c7 is in [789]. => r5c9 is actually in [56], as 8's too high On the right angle, r5c7 is [78] on the GW, and r5c8 is also [23]. Hypotenuese ----------- If r3c4 was high, it could only be 6, and require a 1 on both the GW and the RSL => r3c4 is in [234], and along the GW r3c5 is in [79], and r2c6 is in [12] => to sum with [789] with [12] on the opposite, r1c5 must be [567] => to sum to [6789] with [234]+[567] on the hypotenuese, r2c5 is in [12], but 2 requires another 2, so it's 1 => r2c6 is 2, r3c4 is 3, and r1c5 is 5 => along opposite, r3c7 is 7, r4c8 is 2, and r5c9 is 5, making r6c8 3 and r5c7 8 => along hypotenuese, r4c4 is 9, and r6c3 is 8 => cleaning up the angle GW, r3c5 is 9 => cleaning up box 2, the 4 must be in column 4, so r5c4 is 2 and r5c5 is 4 Adjacent -------- The RSL sums to 5+3+N and to 4+8+M, so N=M+4 From N=[56789], [5_78_] are excluded from r6c7. From M=[12345], [_234_] are excluded from r8c4. => Only 9+3+5=17=5+8+4 is OK, so r8c4 is 5 and r6c7 is 9 => naked 7 in r5c8, leaving a [46] pair in box 6, and a [36] pair in row 5 => naked 9 in r1c9 and naked 2 in r1c7 Column 4 -------- The 8s in boxes 4 and 8 mean that column 4's 8s must be in box 2 => r3c6 is 6, making r1c6 the 7, and leaving a [48] pair => r5c6 is the 3 in row 5 and r5c1 the 6 => r9c5 has become a naked 3 and r8c5 a naked 2 => r6c4 can only be 1, making r9c4 the 6 and r7c4 the 7 => Box 5 completes 8 above 5 r1c3 and r1c8 have now become a [46] pair, making r1c4 the 8, r1c1 the 1 in the row, and r2c4 the 4 completing the column Acute Angle ----------- Recap: we've got a sum to 9 on the hypotenuese, a sum to 17 on the adjacent, and a GW between them. With a shortfall of 8, the elements on the hypoteneuse must be lower than the ones on the adjacent side. r7c2 can't be [234], so is 1. => In order to sum to 8 with r9c1, r8c2 can't be [123457], so is 6, and r9c1 is 2 => The onyl way to get the adjacent to sum to 2+N+M=17 is with r8c3 being the 7, and r9c2 the 8 And you're in speedrun mode by now, I hope... Speedrun -------- => row 2 full resolves 8-7-...-6 => box 3 fully resolves, completing row 1 and boxes 6 and 5 with it. => row 6 completes 7-2-... => column 8 completes .../6/8/9, resolving box 8 => column 7 cmpletes .../5/1/4 => box 7 completes, as do rows 2 and 3