For Turkey Day

Probability Level 1

Can you trace out the lines of the entire turkey?

Things to keep in mind while tracing:

1 ) 1) No picking up your writing or tracing tool.

2 ) 2) No doubling back along a line already traced.

Hint: Careful with the beak!

No, It's Impossible Yes, I can

Zandra Vinegar by Zandra Vinegar

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3 solutions

Zandra Vinegar Staff
Nov 25, 2015

This particular turkey is a special case for this kind of tracing problem.

As you can see in the coloring below, this turkey is made out of 1 beak + 6 overlapping, squiggly loops. Additionally, one loop is special: the green loop intersects every other loop at least once. As a result, here's our strategy:

Since any loop can be traced all at once, ending wherever you began, start at the left side of the beak and move (on purple) to green, then trace along green and, any time you reach a new color, trace that entire colored loop, coming back to the green loop where you left it. Lastly, when you get back to purple, finish purple returning to the diamond where you started and then trace the beak last.

There are many other ways to trace this Turkey of course, but this kind of method will serve you well for any picture that is made almost entirely out of a collection of overlapping loops. Actually, I'm pretty bad at tracing so I basically can't do this puzzle by actually trying to trace it -- too many squiggles. In this case, the mathematical approach is easier in my opinion. ;-)

Happy Thanksgiving!

An example path:

22 Helpful 0 Interesting 0 Brilliant 0 Confused

While I can understand the results described, I would respectully submit that the righthand side of the body of the turkey appears to also have two odd nodes, primarily because of the scallopping of the first layer of feathers (sorry, I cannot detail for some reason, but I'm speaking about the intersection of red and green). The picture has four odd nodes and cannot be traced in a single line

Paul Wanger - 5 years, 6 months ago

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I agree with you, this is why I selected 'No'. The red and green impermissibly overlap on the viewer's right side of the turkey's body. Thus this is impossible.

Greg Wallace - 5 years, 6 months ago

Paul and Greg -- Thanks for noting this ambiguity. I see your point and have updated the image in the post. When I drew the image, I had the lines colored as in the solution and then I made all of the coloring black at once to create the image in the problem. Having thought about it as an intersection for the process of writing up the solution, it didn't catch my attention when I removed the color, but I appreciate that when you look at it with unbiased eyes, that that intersection is unclear.

In the future, if you use the reporting feature when you spot bugs or ambiguities in problems (click the three horizontal line icon to the upper right of the problem post), writers will usually notice and correct their work much faster compared to when a report is posted as a solution/in a solution discussion. Thanks in any case though, I think the change clearly makes this a better problem! :)

All the best (and happy Thanksgiving of course!)

Zandra Vinegar Staff - 5 years, 6 months ago
Alan Guo
Nov 24, 2015

A connected graph is traceable if and only if there are 0 or 2 nodes of odd degree, and the remainder are even. (This is because an even node consists of equal amount of input and output paths, and the odd nodes act as beginning and ending points, and when removed ('used'), it becomes an even node. By connectivity, the graph can be traced.)

The beak consists of two nodes of degree 3, and every other node is even. Therefore, starting at one of the beak nodes, it is possible to trace around.

7 Helpful 0 Interesting 0 Brilliant 0 Confused

Nice work! Can you prove the theorem you're citing: that "a connected graph is traceable if and only if there are 0 or 2 nodes of odd degree?"

Zandra Vinegar Staff - 5 years, 6 months ago

It is impossible. You can't trace out the entire eye and the region around the beak.

Zhi Wei - 5 years, 6 months ago

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It is not impossible, you just have to start at one of the top corners of the beak, where 3 arcs (lines) intersect. See Zandra's diagram above for a method.

Stewart Feasby - 5 years, 6 months ago

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Does not work even if that is the case. If you start from beak, you can't trace the entire purple unless you trace twice.

Zhi Wei - 5 years, 6 months ago

Do the eye loops start and stop and the same point? If that's the case then it's possible, if not then I don't see how you'd go about drawing the eyes

Danny Shiraz - 5 years, 6 months ago

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@Danny Shiraz – "Do the eye loops start and stop and the same point?" -- yes, that is intended to be the case. I lost a little detail in the image when I made my line-thickness wider, but the eyes are two circles tangent to the edge of the head.

Zandra Vinegar Staff - 5 years, 6 months ago
Amrit Nimiyar
Dec 8, 2015

Can we start with the beak? Please tell

0 Helpful 0 Interesting 0 Brilliant 0 Confused

Definitely! (In fact, in order to be successful, you will have to start at one of the two ends of the beak, where 3 lines come together.)

Zandra Vinegar Staff - 5 years, 6 months ago

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