# Riffing off of Dudeney

Attributed to H.E. Dudeney, the following puzzle is possibly the world's most recognized coin rearrangement puzzle. So it is a mystery to me why the two related puzzles that follow can't be found anywhere. This should correct that oversight.  The rules are the same as for Dudeney's original. A move consists of sliding a coin from its current position into a new position into which it fits snugly, i.e., it touches two other coins, which fix its position.

I was lax in my definition of "fits snugly", for which I apologize. If three coins are in a line, none of them fits snugly in its position. Mathematically, the middle coin's position is defined exactly by the positions of the two end coins because there is only one spot at which the middle coin is tangent to both end coins. However, I was going for gut feeling snugness, not mathematical snugness. So, to clarify, coin A fits snugly against coins B and C if the A touches both B and C and the centers of the coins are not colinear.

• "The rules are the same as for Dudeney's original" Which are? – Peter Taylor Sep 6 '18 at 9:32
• What exactly is "sliding"? I know a way of two steps, each slides one coin, that solves those puzzles. There must be some more mystery that I don't know. – puck Sep 6 '18 at 15:39
• @puck, usually, a slide is a continuous move of a single coin, along the table surface, in such a manner that no other coins move. (Also, as OP says in the puzzle, here the slide must end in a spot where the moving coin touches at least two other coins.) – Bass Sep 6 '18 at 17:38
• If you have to lift the edge of the coin, you're not sliding it. – Steve B Sep 6 '18 at 19:42
• Ok that's what I guessed. But I forgot the requirement to place the coin so that it touches two other coins. This makes it more difficult than just two steps. – puck Sep 7 '18 at 4:08

First puzzle in 4 moves:

 . o o
o o o
. .

o o .
o o o
. .

o . .
o o o
o .

o o .
o . o
o .

. o .
o . o
o o

Second puzzle in 5 moves:

   . .
. o o
o o o
.

. o
. o o
o o .
.

. o
. o .
o o .
o

. o
o o .
o . .
o

. o
o . .
o o .
o

o o
o . .
o o .
.
I originally had a 5-move solution for the first puzzle, but it was pointed out that steps 2-5 of this solution for puzzle 2 worked as a 4-move solution for puzzle 1.

The way I solved these is

to work backwards. Each time you have to move a coin that touches two others. This is much easier since near the ending position there are fewer inverse moves available than there are moves available near the starting position.

I do not know if these are optimal. I would not be surprised if shorter solutions are possible.

• Your solution to the second one, which us optimal as far as I can tell, contains a 4 move solution to the first one rotated 60 degrees. – Steve B Sep 7 '18 at 2:02
• @SteveB So it does. I'll edit it. – Jaap Scherphuis Sep 7 '18 at 7:56

The first one is possible in 4 moves:

 . a b
c d e

.
d a b
c   e

c
d a b
. e

c
d   b
. a e

c
d   b
e a


And so is the second one:

   .
a b
c d e

d .
a b
c   e

d a
b
c . e

d a
.
c b e

d a
c
b e


If you are allowed to pick up the coins (instead of just sliding), then the first puzzle can be done in 2 moves:

   a b
c d e
.

a
c d e
. b

a
c   e
d b


I didn't see a 2-move picking-coins-up solution for the other one. Adding a move to the first solution (after mirroring it) gives a simple three-move solution, of course.

• You solved the first the way I did. Your solution of the second follows one interpretation of the rules that my original wording allowed. Your 4th move does not follow the rules as I intended them to be. I will edit the puzzle description to clarify my intention. – Steve B Sep 6 '18 at 19:19

First you move B down between C and D as in the diagram, then slide D out between A and E, then move it in between B and E from the outside. • D won't slide, it's blocked inside the A-B-E triangle. – Bass Sep 6 '18 at 6:55
• It needs to slide? Hmm I didn’t think of that... – Rohit Jose Sep 6 '18 at 6:58
• Wait, I could slide D out through the space between A and E – Rohit Jose Sep 6 '18 at 7:29
• @Rohit Jose That space isn't big enough. Try it with real coins. – Steve B Sep 6 '18 at 19:20
• I see what you mean – Rohit Jose Sep 6 '18 at 19:21