This puzzle is part of the ‘Twelve Labours’ series, but can be solved independently. Previous instalments can be found here: Prologue | 01 | 02 | 03 | 04 | 05 | 06

Emerging from the barbers with a new look that he really hadn’t planned on, Hercules was relieved to see from his to-do list that he was finally halfway through his labours for the day. But time was ticking on...

Hauling his ever-heavier flatbed trolley quickly up the road to Cleats ‘N’ Balls, he was rather out of breath when he walked in through the door. “Cleats” – as it was often known – stocked all kinds of sporting supplies and also doubled as the reception area for the golf course that lay just behind it. It had been run by the same family for generations, the current manager being a young woman called Pasiphaë, who truly loved this job.

“Here are the team kits your mother ordered,” she said with a huge smile, handing Hercules a bundle of shirts, shorts and socks. “The boys will look super smart in those. Can I interest you in a game of golf while you’re here? On the house...”

“No, I’ve got a lot to be getting on with today, thank you,” replied Hercules, shortly.

“Your mother will be disappointed,” said Pasiphaë, with a sly look and a raised eyebrow. “After all – there’s something else she ordered too... and I’ll only tell you what it was if you can beat our golf course.”

Hercules closed his eyes for a moment and breathed deeply, before Pasiphaë gestured towards a map on the wall:

enter image description here

“This is our Herugolf course,” she explained, before pointing out the list of rules on the wall nearby:

Rules of Herugolf: (Adapted from Nikoli)

  1. Move (hit) all circles (balls) one or more times, and bring them to a cell with a red H (hole). Each ball must end up in a different hole.

  2. Show the movement of a ball by an arrow, with the tip of the arrow in the cell where it stops. The arrow cannot cross other balls, holes, or arrows.

  3. When it is first hit, a ball travels across as many cells as the number inside it, in a straight line, either vertically or horizontally (no diagonals, no changing direction partway through a move).

  4. Each subsequent shot follows the same rules but crosses 1 cell fewer than the previous shot (e.g. a ball numbered ‘3’ crosses 3 cells on its first shot, 2 on its second, and 1 on its third). The direction of travel may (but doesn’t have to) change after (not during) a shot. When the next shot length becomes 0, or the ball stops at a hole, the ball cannot move any further.

  5. A ball cannot leave the grid (“out of bounds”) and cannot stop at the end of a shot in a water hazard or sand trap (shaded cells), although it may pass through them if the shot ends on the grass (unshaded cells).

“If you can complete our course you’ll receive a hint as to where you might usually find this 10-letter item – from that you should be able to work out what it is.” Pasiphaë reached behind the counter and snatched up a sheet of paper, which she passed to Hercules. “You’ll also be needing one of these more detailed map printouts to guide you through the course. Good luck!”

TASK: Logically deduce the solution to the Herugolf puzzle (below) and resolve the resulting hint to learn what other item (singular) Hercules’ mother has ordered from Cleats ‘N’ Balls. Please explain your reasoning with diagrams to show the logical steps that led you to the solution.

enter image description here

  • $\begingroup$ must it stop AT a hole after one move? $\endgroup$ – Omega Krypton Nov 15 '19 at 9:34
  • $\begingroup$ @OmegaKrypton It must eventually stop at a hole at the end of A move, but for a 3-ball that could be after 1, 2 or 3 moves, for a 4-ball that could be after 1, 2, 3 or 4 moves, and for a 5-ball that could be after 1, 2, 3, 4 or 5 moves. I recommend clicking the Nikoli link for an example :) $\endgroup$ – Stiv Nov 15 '19 at 9:39
  • $\begingroup$ @OmegaKrypton Out of interest, what's the reason for dropping the herugolf tag? $\endgroup$ – Stiv Nov 16 '19 at 8:43
  • $\begingroup$ usually a grid-puzzle only gets its tag after it has accumulated several puzzles, but this is the first one I've seen. usually we wait for 5+ herugolfs to make a tag... $\endgroup$ – Omega Krypton Nov 16 '19 at 14:05
  • $\begingroup$ @OmegaKrypton I see. Has this been discussed in meta before? $\endgroup$ – Stiv Nov 16 '19 at 16:59

To start:

The 5 near the bottom middle must go right. The 5 near the bottom right must go left, or it will make the hole to its right inaccessible. Then, both of those are pushed up, and both only have one way to make it to a hole.
enter image description here

A few more steps in the center follow from that:

The hole that one ball spiraled around can only be accessed in one way.
The 4 near the middle of row 2 must go left; this pushes the 5 right and gives it only one way to reach a hole.
The hole it reaches also can only be reached by one ball: the 3 near the top middle.

enter image description here

The right side can be completely resolved:

The 5 must go up to not cover a hole. Then it must go right, then there's only one way for it to make it to a hole. Then the top hole must be filled by a 3 hooking around the top, and then the others follow easily.
enter image description here

There's progress on the left side too:

The 4 in row 4 must go left to not cover any holes, and it must go to the bottom hole.

The top hole in the left column can only be reached by the 4 in the top row. Similarly, the hole close to it can only be reached by the 3 directly to its right.
enter image description here

And finishing off the puzzle:

The 3 on the bottom can't go up: it would push the unmoved 3 upwards, trapping the 4 on the left. So it must go right.
The two 4s on the bottom both must go up, and they can easily be resolved.
Now, there's only one way to satisfy the 3 near the lower left, then the 4 on the left, and finally the 3 on the bottom.

enter image description here

Extracting an answer:

enter image description here The unvisited cells spell "SOCCER PITCH VERTICES", in reverse. So the answer is the item found at that location: a CORNER FLAG.

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  • $\begingroup$ Wow, not only did you solve it but you managed to redraw the grid and map out an explanation all in 40 minutes!! That's impressive - well done :) Correct final answer and beautifully illustrated; +1 and a green checkmark incoming! $\endgroup$ – Stiv Nov 15 '19 at 10:13
  • $\begingroup$ @Stiv Thank you! I solved it in 10-15 minutes or so (using the tool at http:puzz.link/p?herugolf, and screenshotting and explaining my progress as I went). An embarrassingly long amount of that time was spent not realizing that the clue was giving the location of an object and not another step. Thanks again for the fun puzzle! $\endgroup$ – Deusovi Nov 15 '19 at 10:25
  • $\begingroup$ Ooh, I didn't realise that tool existed! I'm definitely going to use that in future for the set-up as well as the solve... And you're welcome :) Glad you're appreciating the series! $\endgroup$ – Stiv Nov 15 '19 at 10:27
  • 1
    $\begingroup$ From the first deduction: "The 5 near the bottom right must go left, or it will make the hole to its right inaccessible." But the hole to it's right is connected straight up to the 4. Whether the 5 near the bottom right went up or left had no bearing on the hole on the right. $\endgroup$ – LeppyR64 Nov 19 '19 at 15:04
  • $\begingroup$ @LeppyR64 Whoops, that's completely correct. I missed the 4 there at the very start. I'll redo the logic if I get a chance. $\endgroup$ – Deusovi Nov 19 '19 at 20:02

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