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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 forCutting

Corners JindřichMichalik

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 ofCutting 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 Figure1, 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

Figure1). The player with more areas of their color wins.

In the example in Figure1, 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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