Start with a numbered 6×6 dinner plate. Get the student at the back left chooses any number. (Do not explain the rules in advance. That’s boring.)

The student chooses 14. You say “This is where we are going to start adding our spaghettum. All Italians know that you should never cut your spaghetti.”

 

Ask a student 3 seats away to select the smallest number touching 14. That is – which is smaller 13, 8, 20, or 15? The student replies 8.

“Which is the smallest number now?” You ask the next student three away from the last.

“That’s where our spaghettum will go…” (I love the fake word spaghettum meaning the singular of spaghetti 😉

The child answers 2. Each time a child answers draw a bit more of the spaghetti or make an arrow. Keep on going.

Go to the smallest neighboring number NOT visited before.

 

There is only one place to go from 12…

Go to the smallest neighboring number NOT visited before.

 

“Do you think we will lay down a full-length spaghettum?” You rhetorically ask but do not expect an answer. Just keep going…

Go to the smallest neighboring number NOT visited before.

Starting at 14 you are successful! The spaghettum does not need to be cut!

 

We end at 29 after visiting ALL the squares on our plate! Let’s color 14 green to indicate a success.

 

 

Another student should choose a different number.

 

They choose 31… which does not end well.

Their spaghhetum must be cut short! You can see that once they get to 30 there is nowhere to go. Cutting spaghetti short is a horrible thing for a Italian!

Let’s color 31 red to indicate failure.

 

After students understand successes and failures ask the whole class to collectively explore which spaghettum starting numbers work. In a few slides I will give you the answer. Don’t peek without trying it out and trying to identify patterns.

In my elementary classroom the students engaged well and yet made multiple errors. We entered a discussion on how to limit errors as a group. They ended up suggesting a few solutions including the one we informally adopted “two people should independently figure out the color of a square rather than relying on just one person.”

Before I show you the answer…

You may ask students to fill in a 4×4 plate with the numbers 0 through 15 so that ALL numbers work. Do all the numbers in this example leave the spaghetti uncut?

 

No! – In fact only three work!

Spoiler alert… solution to the 6×6 dinner plate follows…

 

The solution to the 6×6 dinner plate. Two patterns emerge that students should understand: the top row is red except for the ends. The second last column is mostly red.

There are also some confusing anomalies… the 2 and 11.

How will the 8×8 or 10×10 dinner plate look? Students should hypothesize and collectively check.  Spoiler alert…

The solution to the 8×8 dinner plate. The first two patterns we talked about are still there… but some students may start to see other patterns. That’s good. Identification of patterns is the objective.  The reason WHY these new patterns happen is too complex to talk about in front of most students although some will understand WHY but not be able to articulate – and a few will understand and be able to articulate. If the whole class can identify some patterns and predict how the 10×10 dinner plate might look – that is fantastic. Spoiler alert… 10×10 dinner plate coming up…

 

 

How many of their predictions were true?

 

There are many directions you can take this. Here is a dodecahedron. Does it work?

Note to mathematicians: Is there a polynomial time algorithm to figure out if a graph of n nodes can be numbered 1-n such that starting at any node results in a success?

No – this dice fails for 5 and 7. What about a standard twelve sided (dodecahedral) dice? What about a standard twenty sided (icosahedral) dice? Is there any numbering system that works on these dice?

 

 

Another direction is to explore different tiling patterns. Is it possible to have complete success with either of these?

 

No – I think this is as good as you can do with six failures.

 

I’m not as confident that this is the best you can do with this size…

 

What about solving for odd numbered dinner plates? Can this be successful?

 

No – it is much less successful. Notice the checkerboard pattern… that is not a co-incidence.

Spoiler alert… a best possible 4×4 dinner plate is on the next slide…

 

Here is a successful 4×4 dinner plate. One of my students asked if it is possible to get an all red plate?

Spoiler alert… the following slide has a 8×8 solution…

 

Here is a successful 8×8 dinner plate.

 

One last idea… it might be interesting to note where each spaghettum ends up…

Here are the puzzle-sheets. It contains this page of 6×6 dinner plates… a similar page of 10×10 dinner plates, and a page of empty 4×4 dinner plates.