:: Eric Shamblen’s Puzzle Blog ::

KIM asks:

IF YOU HAVE 3 HOUSES AND EACH NEED TO HAVE ELECTRIC, WATER AND GAS CONNECTED, IS IT POSSIBLE TO DO SO WITHOUT CROSSING ANY LINES?

Answer: Given the wording of your question, yes.  One way is to simply run the water lines underneath the houses, like so:

Clearly, at no point do any of the utility lines cross any of the others. You could get the same result by running the electric lines over the houses. Obviously, in the three-dimensional real world, “crossing lines” is not generally a difficult problem: just move one of the lines up or down.

I suspect, however, that you were trying to ask a classic mathematical puzzle, which is stated more precisely as: “Suppose there are three houses on a plane and each needs to be connected to the gas, water, and electric companies, also on the same plane. Is there a way to connect them without any of the lines crossing each other, without using the third dimension, and without allowing the lines to go through the houses?”

Note the difference in the wording, which eliminates my solution above. The part about everything being on the same plane is surprisingly important: if the houses and utilities are arranged on, say, a doughnut-shaped object like a torus, then the puzzle is solvable even with the other restrictions.

If we restrict ourselves to the flat, paper-and-pencil world (or, for that matter, an electronic circuit board, where trying to connect different areas of the board without crossing lines is a common design problem), then no, the puzzle has no solution. This fact was proven in 1930 by Polish mathematician Kazimierz Kuratowski. Stated formally, the complete bipartite graph K_3,3 is nonplanar.

For more information, Wikipedia has a good entry on the topic. Thanks for your question!

I got another email asking for help, this time from LaQuanda:

Engineers’ Week Math & Logic Puzzle #1: Marble Madness

A scientist gave one of four sealed boxes containing red and/or green marbles to the following people; Joe, Bob, Susan, and Kim. There were 3 marbles in each box, and the number of red marbles was different in each one. There was a piece of paper in each container telling which color marbles were in that container, but the papers had been mixed up and were ALL in the wrong containers. He then told all of them to open the box, take 2 marbles out of their box, read the label, and then tell him the color of the third marble.

So Joe took two red marbles out of his box and looked at the label. He was able to tell the color of the third marble immediately.

Bob took 1 red marble and 1 green marble from his box. After looking at his label he was able to tell the color of his remaining marble.

Susan took 2 green marbles from her box. She looked at the label in her box, but could not tell what color the remaining marble was.

Kim, without even looking at her marbles or her label, was able to tell the scientist what color her marbles were.

Can you tell what color marbles Kim had? Can you also tell what color marbles the others had, and what label was in each of their boxes?

CAN SOMEONE HELP WITH THE ANSWER PLS?

This is what is known as a meta-puzzle. Although you are not given all of the information needed — the label in the box — the fact that someone with that information was able to solve it is itself enough information to solve the puzzle. Raymond Smullyan wrote a number of terrific puzzles in this style in his book Alice in Puzzle-Land.

This particular puzzle is not as clearly written as it should be; there appear to be some unstated assumptions. Do the students know what marbles the other students pulled out, and what their responses were? It seems they must, or else Kim would not be able to identify the color of her marbles without looking at her own marbles or slip of paper.  Based on this assumption, we can proceed to solve the puzzle.

SPOILER ALERT: DO NOT READ FURTHER IF YOU WANT TO TRY TO SOLVE THE PUZZLE YOURSELF!

Let’s start with Joe. He pulled out 2 red marbles, so we know the box contained either 3 red marbles or 2 red marbles and 1 green one.  We also know that after he looked at the label, he was able to identify the third marble.  What could the label have said to make Joe certain of his answer?  Suppose the label said “3 green.”  Obviously it’s false, but it would be false regardless of the color of  the third marble.  It’s not enough information to identify the third marble.  Since Joe was able to identify the third marble, this must not have been the label he saw.  Suppose the label said “2 green, 1 red.”  Again, obviously false, but again not enough information.  Suppose it said “2 red, 1 green.”  Then Joe would know the remaining marble had to be red, since the label was false. If the label said “3 red,” Joe would likewise know the remaining marble was green.  But without knowing which of those two labels Joe saw, we can’t tell the color of the third marble. 

Now consider Bob.  He pulled out 1 red and 1 green marble, so he either has 2 red and 1 green or 2 green and 1 red.  He’s already heard Joe’s response; if Joe had said his third marble was green, so that he had 2 red and 1 green, then Bob, knowing he couldn’t have the same pattern as Joe, would know his third marble must be green without having to look at the label. Since he did have to look at the label, we know Joe must have said his third marble was red. So Joe had three red marbles, and his label said “2 red, 1 green.”

Back to Bob. If his label said either “3 red” or “3 green,” he would not have enough information to tell the color of his third marble. His label couldn’t have said “2 red, 1 green,” because Joe’s did. So his label must have said “2 green, 1 red.” Since the label was false, Bob actually must have had 2 red and 1 green. His third marble, then, was red.

On to Susan. She pulled out two green marbles and looked at her label, but still could not figure out the color of her third marble. Suppose her label had said “3 green.” Since she knows the label is false, she could have confidently said her third marble was red. Since she didn’t, that couldn’t have been her label, assuming a reasonable level of intelligence (more on this later). Her label must have said “3 red,” which wouldn’t give her any clues; she could have either 2 green and 1 red or 3 green.

Finally, we have Kim. By process of elimination, she knows her label must say “3 green.” She also knows, like Susan, that she must have either 2 green and 1 red, or 3 green. But since her label must be false, she can’t have 3 green, so she knows she must have 2 green and 1 red — all without looking at her marbles or her label.  Susan, by process of elimination, must have had 3 green.  Puzzle solved!

One final comment:  aside from the assumptions, this puzzle contains a flaw.  There is forced stupidity on Susan’s part.  If Kim could identify what marbles she had based on the information that was already revealed, why couldn’t Susan?  Yes, her label said “3 red,” and with two green marbles she couldn’t tell if she actually had 3 green or 2 green, 1 red.  But she could have applied the same logic Kim did, and deduce Kim’s label and marbles — and then her own, by process of elimination.  So why didn’t she?

 Because then the puzzle would be unsolvable.  If Susan had been able to identify her third marble, we could not have been able to tell whether it was because she had a “3 green” label or if she had performed the logical steps described above.  In order for us to be able to solve the puzzle, Susan had to be smart, but not too smart — at least, not as smart as Kim.

So I got this email the other day:

Hi. I got this puzzle from a company called Skor-Mor Corp. back in 1971. The idea is to take the string/ring/block assembly off of the wire loop. It took me about 45 min. of fiddling to solve it and then I was able to demonstrate the solution — even behind my back! I put the puzzle away for about 8 years, forgot the solution, and it’s bedeviled me ever since. I’ve tried without success to locate either a picture of the same puzzle or a solution. Skor-Mor Corp. no longer exists. Here’s hoping you might know something. — Markham

A mystery! The game was afoot!

First I Googled “skor-mor.” The first result was Skor-Mor Online. This was too easy… oh. “Skor-Mor is one of the nation’s largest bowling equipment suppliers…” It WAS too easy. Scanning the next few results showed more bowling references mixed in with some puzzle references. Better narrow the pool.

Next, Googling “skor-mor puzzle.” Some interesting links, including one back to my own blog. Well, if I can’t find the information on my own, perhaps Mike Reilly still has some connections. In any case, there are 181 results, which is still more than I want to search through.

What else do I know about this mysterious puzzle, other than the manufacturer? The goal is to remove a ring assembly from a wooden block… Googling “skor-mor puzzle rings.”

Aha. First result is Rob’s Puzzle Page - Other. The description is: This puzzle is called Right On and was produced by Skor Mor. …. This “Magic Ring” trick consists of a fairly long loop of chain and a metal ring. … Definitely worth further investigation.

The site turns out to be a long, fascinating history of various sorts of puzzles throughout modern history. Skor-Mor is mentioned 4 times as a puzzle and game manufacturer, but not the particular type of puzzle I’m looking for. A brief look at the menu on the top of the page shows links to other categories of puzzles, but again, nothing that describes this particular type. Back to the search results.

A few more false starts on page one, and then on the second page I start seeing results for “SKOR-MOR’S STRINGS’N'RINGS WOOD” on E-bay and similar sites. Looks promising, but apparently these are old listings — the product is no longer actually listed when I go to the site.

Let’s try “skor-mor strings rings.” Second result is Strings N Rings Description: “This nicely crafted set was produced by Skor-Mor (part of their “Thinker” series). Each consists of strings and rings (of course). … “ Looks good. Clicking on the link, I see 4 pictures, including this one:

Ring-A-Ling Puzzle
Remove the center ring (and of course place it back).
The description sounds wrong, but this looks pretty much like Markham’s puzzle, ignoring the large wooden block apparently used for display purposes. Googling “skor-mor ring-a-ling.” Only two results: the page I was just on, and Rob’s Puzzle Page again, only this result says Rob’s Puzzle Page - Tanglements. Description: I got a partially filled wholesale box of old Skor-Mor Thinker Series puzzles. … 1x F18 Ring-A-Ling (equal to Left Brain), 1x (no label, … Clicking on the link, I see that Rob apparently does not believe in consistent internal navigation. This page has a totally different menu from the first page I saw, which did not link to this page. But searching the page itself for “ring-a-ling” I find this:


It’s quite obviously Markham’s puzzle, but my goal is to find not just the name of the puzzle or a picture of it, but the solution, if I can. And Rob’s Puzzle Page doesn’t have the solution. Moving on…But wait a minute. Something looks wrong. I look back at Markham’s original picture, and it hits me. There is a clear difference between the picture on Rob’s Puzzle Page and the one Markham sent me, one that was not so obvious in that first result I found. Markham’s puzzle only has two rings, but Rob’s has three. And what was the description on that first result I found? “Remove the center ring (and of course place it back).”

There is the answer: Twenty-odd years ago, Markham successfully removed the center ring, but did not put it back! Years later, when he came across the puzzle again, he had forgotten that there ever was another ring, but remembered that the goal of the puzzle was to remove something from something else. So he made up an explanation. “The idea is to take the string/ring/block assembly off of the wire loop” — which is impossible. No wonder he couldn’t solve it!

The only thing left to do was to tell Markham my findings, but my emails to him kept getting blocked as spam. Well, maybe he’ll read my blog…