# Is this Hashi puzzle unsolvable?

I found this puzzle in my dad's bathroom. The book is "Things To Do While You Poo On The Loo". I think the puzzle is malformed. Here is a Penpa link to an online solver. Is there actually an answer to this puzzle?

Hashi Puzzle

This is a single player game. Each puzzle is based on a rectangular arrangement of islands where the number in each island denotes how many bridges are connected to it. The object is to connect all islands according to the number of bridges while obeying these rules:

• No more than two bridges can connect any two islands.
• Bridges can only be vertical or horizontal (not diagonal)
• Bridges cannot cross islands or other bridges.
• When completed, all bridges should be interconnected enabling passage from island to island.
• It may be possible. I see nothing that prevents you connecting bridges from an island to other bridges rather than always pairing islands. Commented Jun 17, 2023 at 21:40
• It's a thought, but it doesn't conform to the rules of a standard Hashi puzzle. Commented Jun 19, 2023 at 9:46
• Is it possible your dad has played a rather clever prank on you? Commented Jun 19, 2023 at 14:06
• I don't think so. The puzzle, for interest's sake, was found in a book called "Things To Do While You Poo On The Loo". It's Amazon print-on-demand shash, not a massively surprisingly place to find something like this. I could see the author doing it for a prank. Commented Jun 19, 2023 at 14:27
• FWIW, this game is available as part of Simon Tatham's Portable Puzzle Collection (available for most devices) but is called "Bridges". Its original Japanese name is "Hashiwokakero". Commented Jun 19, 2023 at 16:22

There is no solution (see diagram).

First, the 8-island must be connected as indicated.
Second, the 1-island in the top row must be connected as indicated.
Third, the 3-island in the top row must be connected downward as indicated.
Fourth, the 1-island in the bottom row (5th column) can’t be connected to the 2-island to its left because that 2-island already has enough bridges.
Therefore the 1-island in the bottom row (5th column) must be connected as indicated.
Lastly, the 2-island in the bottom-right corner currently has 1 bridge attached to it. It needs one more bridge but the island to its left and the island above it already have enough bridges. Therefore no solution is possible.

• Yep, the right 4 should've been a 5, and it will have a unique solution. Commented Jun 17, 2023 at 12:57
• Or the top 1 could have been a 2. Commented Jun 17, 2023 at 22:09
• Single number changes to fix this one: top 1 could be 2, top 3 could be 2, right 4 could be 5, 8 could be 7. We cannot change bottom right 2 to 1 as there is no path over all islands. I believe that should be it - the problem we are dealing with is that there is one extra bridge for the right 4 and these are all the options to remove it (or allow it). Commented Jun 19, 2023 at 10:42

Here is another argument for why this puzzle is unsolvable.

The Handshaking Lemma is the simple but powerful idea that each edge uses up a value of 2 (one from each end), so the total amount must be a multiple of 2, i.e. even. In this puzzle the total amount is odd, since there are an odd number of odd digits in the puzzle - three 1's, three 3's and one 5, so seven odd digits.

• Apart from its simplicity, this proof is nice because it also proves that the problem is unsolvable even if you don't restrict the edges to be horizontal/vertical! Commented Jun 18, 2023 at 7:12
• This is really elegant, I wish I could pick a second accepted answer. The first one works through it nicely, but this is a very clear mathematical proof that I really enjoy. Nice. Commented Jun 19, 2023 at 9:47
• @BlueRaja-DannyPflughoeft ... or you allow them to cross, or you allow more than two parallel edges, or you don't require everything to be connected. Commented Jun 19, 2023 at 14:01