There are two heuristics that cover almost all of this problem. When I say "[square] can only be served by the [number] (at [numbered square])," I mean that there is only one numbered square in its row or column that can have a line leading to it without crossing another line or exceeding its value. And when I say that "[numbered square] must extend to [square]," I mean that if the numbered square did not have a line going at least as far as the indicated square, it would not be able to achieve its value. I am using yellow highlights to indicate a square whose value has been achieved.
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JW must extend to JU. KV can only be served by the 3, and LS can only be served by the 8. IU must extend to GU. GW must be served by the 8. IU must extend to EU, IT, and IV. HV must extend to FV. GR can go at most two north, three west, and two east, so it must go south at least to IR. HS can be served only by the 7.FT must extend to ET and HT. GS must extend to FR, GP, and KR. KS can only be served by the 3. JS and KW can only be served by the 8. HX can only be served by the 8 at EX.
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JQ must extend to JP. IP can only be served by the 6. KP must extend to KN. LN must extend to LQ. KP must extend to KP and KQ. LM can be served only by the 4. JQ must extend to JN and IQ. IM must extend to GM. HN and HP can only be served by the 6. HO must extend to EO. FN can only be served by the 5. DM must extend to BM and EM. FQ can only be served by the 9. DP can only be served by the 3. EP must extend to CP. GR must extend to ER and LR. LX must extend to IX. EX must extend to BX and EW. CN must extend to BN. BQ must extend to BS. CR can only be served by the 7. CS must extend to CT. DT can be served only by th 4. DU must extend to CU. CW must extend to BW. |
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EX must extend to either AX or EV, so AV cannot extend to both. So AV must extend to AU. DU extends to DV. EV can only be served by the 8. AX can only be served by the 6. CW must extend to CV and DW. DU must extend to BU. BV can only be served by the 6. AV must extend to AS. AR can only be served by the 5. BQ must extend to BO and BT. AO must extend to AM. DM must extend to FM. CN must extend to CO. EO can only be served by the 6. HQ can only be served by the 6 at JQ. IM must extend to JM. |
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The numbers serving the diagonal squares are 4 4 5 6 5 9 7 3 4 3 3 6. |
Notes: This problem probably took me close to a half-hour to solve. Much harder than the sample problem given, although quite a bit easier than the difficult examples in Rubik's Games, where this puzzle is prominently featured.