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K-12 Unsolved:

Graceful & Mutant Math

A “Not-so-Basic Fact” Theme

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101 And You’re Done

(A Game of Place Value)

Number of Players: 2-4

Materials: one die, paper/pencil to record score

Goal: To get a sum as close to 100 without going over. Get 101? You’re done.

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  1. Take turns rolling a six-sided die. After each roll, decide whether to leave the number as is or multiply it by 10 (e.g. 3 or 30) before adding it to your score.
  2. Each player rolls exactly six times.
  3. Find the sum of your six rolls and determine the winner.

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Creature Curiosities

Goal:

Place consecutive odd numbers (1, 3, 5, 7, 9, 11, 13) into the circles so that the differences of all connected numbers are different.

*This is a 7-node creature.

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Possible Creatures

Not Possible

6-node creature

5-node creature

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What do you NOTICE?

What do you WONDER/CONJECTURE?

If you have a loop, you have more line segments than differences…so it’s impossible because #nodes =# differences and you’d have to reuse them.

Do the extremes have to be connected?

Would this work with even number nodes?

Does it matter what is on the end of the nodes of a snake?

Can you make any n-node snake?

Are there any non-loop creatures that are impossible?

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Graceful Tree Conjecture

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Math Pickle Youtube Explanation

Numberphile Explanation

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Creature Curiosities

GOAL:

Place consecutive odd numbers (1, 3, 5, 7, 9, 11, 13…) into the circles so that the differences of all connected numbers are DIFFERENT.

Are all creatures possible?

Are some creatures impossible?

What do you notice? What makes you curious?

Make your own creature!

Think of a tree-like structure made of circles (we'll call them "nodes") connected by lines (or "edges"). This puzzle is called the “Graceful Tree Conjecture”, first proposed in 1967, and is unsolved to this day!

The name "Graceful Tree Conjecture" comes from the type of structures it focuses on. In this context, a "tree" is a network of nodes and edges where:

Everything is Connected: You can travel from any node to any other node through the connecting lines.

No Loops Allowed: There's only one path to get from one node to another. Imagine a real tree – you wouldn't expect branches to loop back on themselves!

Examples: NON-examples: