The Winning Move for Cutting Corners

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The Mathematical Intelligencer ⚫ © The Author(s), under exclusive licence to Springer Science+Business Media, LLC, part of Springer Nature 2022
https://doi.org/10.1007/s00283-022-10215-9
1
The Winning Move forCutting
Corners JindřichMichalik
In 2019, Jim Henle published an article [1] in which he
introduced a quadruple of intricate games from Sid
Sackson’s book A Gamut Of Games [5] and posed some
interesting mathematical problems. After describing
a winning strategy for the neat game Hold That Line
in [3], we have focused our attention on the perplex-
ing game Cutting Corners. While a winning strategy for the
former game was found by guessing one “lucky” move and
subsequent case-by-case analysis, Cutting Corners seems
too complex to be analyzed without a computer. Our pro-
gram [4] written in Mathematica revealed which of the two
players has a winning strategy, as well as a few interesting
facts as a bonus. The goal of this article is to describe these
results, as well as a few observations regarding the game’s
rules. Besides the original six-move game by Sid Sackson,
we also investigate some variants in which the game ends
after four, eight, or ten moves.
The Rules ofCutting Corners
The game board for Cutting Corners is represented by a
square with two blue sides (rst player’s color) and two red
sides (second player’s color). The sides of the same color
are adjacent to each other; see Figure1, upper left. Players
draw L-shaped lines of their color. The line must start and
end on a side of the square and obey the following rules:
1. A new line must touch a side of the opposite color or
intersect a line of the opposite color.
2. The number of intersection points of a new line must be
equal to the number of lines previously drawn, i.e., the
nth line must cross the previous lines
n−1
times (it is
not necessary to cross each line exactly once).
After six moves, each region with more blue sides than red
is colored blue, and each region with more red sides than
blue is colored red (see the bottom right-hand picture in
Figure1). The player with more areas of their color wins.
In the example in Figure1, Blue wins 9 to 7 (this example
appears in Sackson’s book [5]).
Observations
Let us explore the meaning of rules 1 and 2 above. More
precisely, what are the implications of rule 1? What would
happen if we decided to relax it while adhering to rule 2? Let
us take a look from the perspective of the second player (Red)
rst:
(a) When Red makes their 2
k
th move, then according to rule
2, the new line must have
2k−1
intersection points with
the previously drawn lines.
(b) The number of previously drawn lines is
k−1
red and
k
blue (the new line is red).
(c) If Red wants to violate rule 1, the current move needs
to cross red lines only.
(d) Two red lines intersect in at most two points.
These observations imply that the maximum number of
intersection points for a new red line when rule 1 is violated
and rule 2 is followed is
2(k−1)=2k−2
. According to
observation (a), we conclude that Red cannot violate rule 1
if they follow rule 2 throughout the game. In other words,
the game does not change if we omit rule 1 for Red.
Let us switch to the rst player’s (Blue) point of view:
(a) When Blue makes move number
2k+1
, then according
to rule 2, the new line must have 2
k
intersection points
with the previously drawn lines.
(b) The number of previously drawn lines is
k
red and
k
blue (the new line is blue).
(c) If Blue wants to violate rule 1, they need to cross blue
lines only (in the current move).
(d) Two blue lines intersect in at most two points.
These observations imply that the maximum number of inter-
section points for a new blue line when rule 1 is violated and
rule 2 is followed is 2k. This number can be reached if and
only if all previously drawn blue lines have two intersection
points with the new line. Moreover, the new line must touch
the blue sides of the square, for otherwise, rule 1 would not
be violated. Therefore, all the previously drawn blue lines
must touch the red sides of the square; otherwise, two inter-
section points are impossible. The conclusion is that if ever
Blue draws a line that does not touch both red sides of the
square, rule 1 can be dropped for the rest of the game.
The Program
In this section we provide a brief description of the
program that we used to analyze the game. We begin by
describing the essential data structures and then proceed to
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