Recall that in Sprouts we define a "portal" to be a spot that
connects two regions. It has two "sites", one in each region.
I will write D(k) for a "Dawson's site of depth k". D(1) is
defined as a site of a portal spot whose other site is in a dead
region. D(k+1) is defined to be a site of a portal spot whose other
site is accessible only to a D(k). We can also define a D(0) to
be a dead or nonexistent spot.
We can construct a D(k) from a D(k-1) and a degree-0 spot X with
the two moves:
Y : X-X > ; -- connect X to itself without
separating any boundaries. The new
spot is called Y.
D(k) : X-Y > D(k-1) ; -- connect X to Y, separating the D(k-1)
from the rest of the region. The
outer site of the new spot is a
Given D(k) in an otherwise empty region, the only possible moves
are to connect two of its portal points, leaving isolated D(i) and
D(k-i-2) positions, for 0 <= i <= k-2. This is clearly equivalent to
Dawson's Kayles played with k pins.
Another use of Dawson's sites is to generate positions that tend
to have reasonably large game complexity, since Dawson's Kayles
produces nim-values up to *9.
The following moves will generate a boundary with sites D(i),
D(j), D(k) in clockwise order. Assume a region with D(i-1), D(j-1),
D(k-1), and three degree-zero spots X, Y, and Z.
n : X-X > ; -- connect X to itself without
separating any boundaries.
n+1 : n-Y ; -- connect the new spot to Y.
n+2 : Y-Z ; -- connect Y to Z
n+3 : n+1.X - n+2.Y > D(k-1) -- connect n+1 to n+2, enclosing
D(k-1) in a region bounded by
Y, n+1, n+3, n+2 in clockwise
n+4 : n+3-Z > D(j-1) -- connect n+3 to Z, enclosing D(j-1)
in a region bounded by Z, n+2, n+3,
n+4 in clockwise order.
D(i) : X-n+4 > D(i-1) -- enclose D(i-1), creating a new
After these moves, spot Z is a D(j) site and spot Y is a D(k) site.
If we call such a boundary a D(i,j,k), then my program (a CGSuite
script) tells me that
1. A position consisting of D(2,3,5);D(1,2,6) has nim-value *11,
2. A position consisting of D(2,3,5);D(1,6,2) has nim-value *5.
where ";" means to put boundaries in the same region.
These positions can be constructed from 19 initial spots in 38 moves.
If Aunt Bee can confirm or refute these nim-values, I'd appreciate it
greatly. My program has not been seriously debugged, and has many
possibilities for error.
While I would not be surprised to find a bug, I would be quite
surprised if there were no counterexample among the D(i,j,k);D(l,m,n)
Let D'(k) stand for a position containing only a D(k) site in an
otherwise empty region. Aunt Beast calculates the following
D'(0) = (,;) = *0
D'(1) = (2,;) = *0
D'(2) = (2,a,;a,;) = *1
D'(3) = (2,b,;b,a,;a,;) = *1
D'(4) = (2,c,;c,b,;b,a,;a,;) = *2
D'(5) = (2,d,;d,c,;c,b,;b,a,;a,;) = *0
D'(6) = (2,e,;e,d,;d,c,;c,b,;b,a,;a,;) = *3
Thus, Aunt Beast finds that D'(n) has the same nim-value as a row of n
pins in Dawson's Kayles (at least for 0 <= n <= 6); this suggests that
I have faithfully represented Dan Hoey's D(n) construction in AJS
notation and that Aunt Beast is correctly evaluating the resulting
In AJS notation, a D(i,j,k) boundary is then a boundary (abc) where
a,b,c are D(i), D(j), and D(k) sites respectively. For example, in the
position (2ab,;2,a,;2,z,;z,b,;), the boundary (2ab) is a D(1,2,3)
boundary. According to Aunt Beast, this position has a normal-play
nim-value of *3.
To check Dan's examples, let Q stand for the following partial
Note that in a component containing Q, any occurrences of the spots
a,b,c,d,e outside of Q will be D(2), D(3), D(5), D(1), and D(6) sites
1. (abc,2de,;Q) is a position consisting of D(2,3,5);D(1,2,6).
2. (abc,2ed,;Q) is a position consisting of D(2,3,5);D(1,6,2).
According to Aunt Beast, both positions have nim-value *0.