# Week 19 - Solution: Power Up! (Puzzle 1 of 4 in this series)

When you took a careful look at the “correct” trees (also called “graceful trees”--more on that later!), you saw that no numbers repeated on either the wires or the nodes. All the “X” marks pointed to repeating numbers, and those didn’t exist in the trees with the check marks. Looking closely for a pattern, you also saw that the trees on the left subtracted node from node to get the number on the wire. Cool!

Applying these same patterns to the Leaf Color Changing Machine’s power source, you were able to figure out which node was “1” by subtracting the 7 wire from the 8 node. You were also able to solve for the “4” wire by subtracting the 4 node from the 8 node. This opened up the top branches of the tree. By keeping track of which nodes and wires you'd used, you figured out that the “5” node couldn’t go on the right because there was already a “4” wire below. You tried a “3” and it worked, so you tried some other numbers until you found a home for every node and wire. Nice work! Well done! You’ve restored power to the machine!

(As an aside, Merneith comes back in and tells you that Mentor Zenon, the Silverquicken Math & Science teacher, has sent you a note about this puzzle. It reads, “This is a *graceful tree*, a graph in which no two vertices (nodes) share a label, and each edge (wire) is uniquely identified by the absolute difference between its endpoints. In graph theory, the Graceful Tree Conjecture hypothesizes that all trees are graceful, but we don’t know for sure--that’s why it’s still just a conjecture, or guess.”) Interesting...

With power now flowing to the machine, a monitor springs to life! Four wavy lines cross the monitor and you are perplexed. You look at Merneith who shrugs and says, “cocoon and butterfly!” but does tell you that the number at the bottom of your power source (the number “2”) will be relevant as you continue to try and fix the Leaf Color Changing Machine.

Hmmm, what will you have to do next???