# CAMP 2023 Day 3

This morning, the CAMPers gathered in the auditorium for a collection of math puzzles and the new problem of the day: How many Tower of Hanoi board states are there? The CAMPers had fun trying to work it out, rewriting the problem on the board – in the Greek alphabet. Right before breaking up into their usual Math groups, Japheth led them all in the International Math Salute.

Following up on the argument they made yesterday, the COS group worked together to prove that there couldn’t be an 8th symbol in a 3-Spot-It! deck. They then drew up a graph that brings to mind the “Deathly Hallows” from Harry Potter, showing the 7 cards as nodes and the symbol in common between two cards as edges. Next, they broke up into smaller groups to work on designing a 4-Spot-It! deck – What’s the greatest number of cards we can get with 4 symbols on each card? Some of the CAMPers observed that for the 2- , 3- , and 4-Spot-It! cases, the number of symbols was the same as the number of cards. To look for more patterns, they took the difference between the number of cards in n-Spot-It! and (n-1)-Spot-It! , then took the difference between the differences – getting 2 every time. Leaving this conjecture as a cliffhanger for a later date, the CAMPers embarked on a final investigation – to make one pile for each symbol using existing Spot It! decks.

Meanwhile, the SINE group used tree diagrams to explore the main concepts of combinatorics. Taking the letters S, T, R, and E, they mapped out all the possibilities of what letter could come after each letter in a word, branches branching off into other branches. They followed different branches to make different words like SET, REST, STREET, TEETERS, and – appropriately – TREES. The CAMPers continued their discussion by breaking into two smaller groups to work on these examples: choosing ice cream flavors for a multi-scoop cone, and choosing paths to take when going from Point A to Point B. How many combinations are there? How many choices do we have?

In Computer Science class, the SEC group learned about the random() function, void setup() and void() , variable assignment – how to turn x into (x+1) in order to move a shape somewhere else on the canvas. Their challenge for today: Rather than just moving your drawing around, can you make your drawing move in an animation?

After class, the students joined their elective groups: more magic tricks, more paper puzzles (AKA the Harry Potter debate club), as well as origami.

During lunch, the rain started pouring outside Kline, and the CAMPers had to break out their umbrellas to make it to their next classes – Art and Computer Science.

CSC went to Computer Science, where they continued to learn about truth tables. Then they broke into small groups, using logical operators to write out a combination of A and B that was equivalent to the XOR (exclusive or) operator. Then they took out Little Bits (batteries, cables, and LEDs) to make an XOR statement in real life.

In the Art room, music was blasting while the SEC group continued working on their Spot It! cards, then dove into a variety of artistic and mathematical endeavors – Fibonacci challenges, 4×4 tic-tac-toe, pink paper airplanes, and optical illusions.

Afterwards, the two groups converged in the auditorium – finally dry and ready for some math games.

New ideas, new methods, new questions and answers filled Day 3 with color, excitement, and fun! The CAMPers made so many new discoveries today, leading to even more unanswered questions – perfect to tackle with zeal on Day 4.