# Can this particular game be beaten **URL:** \ **Category:** Factual Questions\ **Created:** [December 22, 2011, 1:37am UTC](https://boards.straightdope.com/t/can-this-particular-game-be-beaten/607025 "2011-12-22T01:37:26Z")\ **Posts on this page:** 13\ **Page:** 1
**Author:** ![Wesley\_Clark](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/wesley_clark/32/20581_2.png) [@Wesley\_Clark](https://boards.straightdope.com/u/Wesley_Clark)\ **Post date:** [December 22, 2011, 1:37am UTC](https://boards.straightdope.com/t/can-this-particular-game-be-beaten/607025/1 "2011-12-22T01:37:26Z")
I don’t know the name of it, but you draw 3 circles and 3 squares on a sheet of paper in any design you want. Then you have a draw a line connecting each circle to all 3 squares, but none of the lines you draw (they do not need to be straight lines) can cross each other. Everytime I and other have tried to play it you always end up having one line that you can’t get because it overlaps another line. So is there a solution to that game, or a name for it? Is there some mathematical principle behind it since you always end up one line short? ---
**Author:** ![yabob](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/yabob/32/2821_2.png) [@yabob](https://boards.straightdope.com/u/yabob)\ **Post date:** [December 22, 2011, 1:43am UTC](https://boards.straightdope.com/t/can-this-particular-game-be-beaten/607025/2 "2011-12-22T01:43:14Z")
No, it’s not possible. Also called the “gas, water, electric” problem. Graph theory - the K3,3 graph is not embeddable in the plane. ---
**Author:** ![yabob](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/yabob/32/2821_2.png) [@yabob](https://boards.straightdope.com/u/yabob)\ **Post date:** [December 22, 2011, 2:09am UTC](https://boards.straightdope.com/t/can-this-particular-game-be-beaten/607025/3 "2011-12-22T02:09:22Z")
Oh, and wiki has an article on it: > **[Three utilities problem](https://en.wikipedia.org/wiki/Water,_gas,_and_electricity)** > > The classical mathematical puzzle known as the three utilities problem or sometimes water, gas and electricity asks for non-crossing connections to be drawn between three houses and three utility companies in the plane. When posing it in the early 20th century, Henry Dudeney wrote that it was already an old problem. It is an impossible puzzle: it is not possible to connect all nine lines without crossing. Versions of the problem on nonplanar surfaces such as a torus or Möbius strip, or that all... ---
**Author:** ![Little\_Nemo](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/little_nemo/32/3120_2.png) [@Little\_Nemo](https://boards.straightdope.com/u/Little_Nemo)\ **Post date:** [December 22, 2011, 2:17am UTC](https://boards.straightdope.com/t/can-this-particular-game-be-beaten/607025/4 "2011-12-22T02:17:27Z")
There’s a cheat. You can do it by drawing a connecting line through one of the circles or squares. ---
**Author:** ![Robot\_Arm](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/robot_arm/32/18280_2.png) [@Robot\_Arm](https://boards.straightdope.com/u/Robot_Arm)\ **Post date:** [December 22, 2011, 2:18am UTC](https://boards.straightdope.com/t/can-this-particular-game-be-beaten/607025/5 "2011-12-22T02:18:11Z")
I think it can be beaten if you roll the paper into a cylinder. ---
**Author:** ![yabob](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/yabob/32/2821_2.png) [@yabob](https://boards.straightdope.com/u/yabob)\ **Post date:** [December 22, 2011, 2:48am UTC](https://boards.straightdope.com/t/can-this-particular-game-be-beaten/607025/6 "2011-12-22T02:48:49Z")
> [@Robot\_Arm](#): > > I think it can be beaten if you roll the paper into a cylinder. Bend it around to form a torus, and you got it. ---
**Author:** ![Rhythmdvl](https://avatars.discourse-cdn.com/v4/letter/r/85f322/32.png) [@Rhythmdvl](https://boards.straightdope.com/u/Rhythmdvl)\ **Post date:** [December 22, 2011, 2:51am UTC](https://boards.straightdope.com/t/can-this-particular-game-be-beaten/607025/7 "2011-12-22T02:51:35Z")
What about a Möbius strip? ---
**Author:** ![yabob](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/yabob/32/2821_2.png) [@yabob](https://boards.straightdope.com/u/yabob)\ **Post date:** [December 22, 2011, 2:57am UTC](https://boards.straightdope.com/t/can-this-particular-game-be-beaten/607025/8 "2011-12-22T02:57:08Z")
> [@Rhythmdvl](#): > > What about a Möbius strip? That will work, too. ---
**Author:** ![ultrafilter](https://avatars.discourse-cdn.com/v4/letter/u/3d9bf3/32.png) [@ultrafilter](https://boards.straightdope.com/u/ultrafilter)\ **Post date:** [December 22, 2011, 3:15am UTC](https://boards.straightdope.com/t/can-this-particular-game-be-beaten/607025/9 "2011-12-22T03:15:01Z")
And a sphere as well. ---
**Author:** ![yabob](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/yabob/32/2821_2.png) [@yabob](https://boards.straightdope.com/u/yabob)\ **Post date:** [December 22, 2011, 3:20am UTC](https://boards.straightdope.com/t/can-this-particular-game-be-beaten/607025/10 "2011-12-22T03:20:21Z")
No, embedability on a sphere is equivalent to a plane. ---
**Author:** ![ultrafilter](https://avatars.discourse-cdn.com/v4/letter/u/3d9bf3/32.png) [@ultrafilter](https://boards.straightdope.com/u/ultrafilter)\ **Post date:** [December 22, 2011, 3:24am UTC](https://boards.straightdope.com/t/can-this-particular-game-be-beaten/607025/11 "2011-12-22T03:24:53Z")
:smack: I blame all the video games played on toroidal worlds. ---
**Author:** ![BigT](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/bigt/32/12044_2.png) [@BigT](https://boards.straightdope.com/u/BigT)\ **Post date:** [December 23, 2011, 4:42am UTC](https://boards.straightdope.com/t/can-this-particular-game-be-beaten/607025/12 "2011-12-23T04:42:44Z")
> [@Little\_Nemo](#): > > There’s a cheat. You can do it by drawing a connecting line through one of the circles or squares. That’s not a cheat. It’s the answer. The problem always forgets that the math only works with points. ---
**Author:** ![Indistinguishable](https://avatars.discourse-cdn.com/v4/letter/i/90ced4/32.png) [@Indistinguishable](https://boards.straightdope.com/u/Indistinguishable)\ **Post date:** [December 23, 2011, 5:30am UTC](https://boards.straightdope.com/t/can-this-particular-game-be-beaten/607025/13 "2011-12-23T05:30:54Z")
A simple proof that it can’t be done on a sphere (and, thus not on a piece of paper, or such things), in the sense in which it can’t be done: First, pick six points on the sphere, call half of them circle-points and the other half square-points, and draw any loop you like that goes through all of the points, comprising six of the desired connections. So long as the loop doesn’t intersect itself, it splits the sphere into precisely two regions. There are three more connections to be made, and each time you add them, so long as you avoid intersecting other connections, you will split precisely one region into two. Thus, after you finish, if you’ve avoided intersections, there will be five regions. But, as each of the 9 connections will border precisely 2 of the 5 regions, this means the average number of connections comprising the border of a region will be 9 \* 2 / 5 = 3.6. This means there would have to be some region with less than 4 connections in its border. But that’s not possible! A border is a loop, and thus must keep alternating between the circle-points and the square-points; thus, it must be made of an even number of connections. And you can’t have just 2 connections in a loop (the second would just be the first in reverse…). So the smallest border possible is made out of 4 connections. Thus, it is not possible to avoid intersections when connecting three circle-points to three square-points a la the OP on the surface of the Earth.