Project Description
In the week of September Fifth to September Ninth, we were assigned various tasks in class, such as finding a solution to a problem relating to finding the smallest amount of squares we could fit inside an 11x13 rectangle, by tiling it with the least amount of squares' possible. The second was Squares to Stairs in which we had to figure out the pattern in squares which would increase in a stair formation each time. The third was the Hailstone Sequence (See Extension) which I will explain further in the bottom, and the last was the Painted Cubes in which we envisioned a 3 by 3 by 3 cube (made of smaller 1 by 1 by 1 cubes) being dropped in paint and then figured out which cubes had how many sides painted. We also watched videos relating to self confidence, synapses in the brain, and how they all correlate to our work, which grew to increase my self confidence by understanding that making mistakes helps my brains grow, not shrink. The videos that stood out to me the most were the ones describing that working through a problem is more beneficial to your knowledge than memorizing an answer or giving up. I definitely like the effort I put into this weeks work, I feel I did far more effort into solving the problems than I had last year or the previous year, something I am sure will reflect into the rest of the year. I chose to cover the Hailstone Sequence as it was the one I had the most fun with, as well as the fact that the German scientist who posed the sequence in 1937 offered a 500 US Dollar reward for solving it, which has still not been claimed. Below, I will go over more detail regarding the Hailstone Sequence.
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Hailstone Sequence
The RulesThe name of the Hailstone Sequence comes from the fact that once graphed, often it looks like the pattern a hailstone goes through for formation - repeatedly rising until it is too heavy to go up again.
The rules of the Hailstone Sequence we were assigned in class are very simple: Start with any positive integer (an initial seed) and obtain a sequence of numbers by following these rules. 1. If the current number is even, divide it by two; else if it is odd, multiply it by three and add one. 2. Repeat, until you cannot go any further. |
StepsThe steps regarding the process are very simple - pick any number. For this showcase, I've decided to bring several examples:
Let's try a few to see the end result. 3; 10, 5, 16, 8, 4, 2, 1, 4, 2, 1, 4; 2, 1, 4, 2, 1, 5; 16, 8, 4, 2, 1, 4, 2, 1, The pattern that becomes obvious is that at the end of the number, it gets caught in an infinite loop of 4,2,1- no matter how many times one tries to change the starting number, the rules make it so that four, two, and one are the end result. |
Problem Extension
"Extend the work on that problem beyond what we covered in class. For example, for Hailstone Sequences, you could change the two rules and explore the new sequences that are created."
This is the assignment that was given to us for the Extension, so therefore, I will change the rules of the Hailstone Sequence.
The new problem I generated: if the number is even, divide it by 4 and add 1, and if the number is odd, multiply by 2 then add 2. These were the sequences I generated.
Here are some examples of how it turned out
16; 5, 12, 4, 10, 3.5
20; 6, 2.5
Unfortunately, I noticed in all cases that this sequence would eventually end in decimals - I tried many combinations, and none ever challenged that theory, similar to how the original sequence always ended in 4, 2, 1.
I really liked the extension of the problem though, as it challenged our thinking more than just explaining the problem - we had to mold the problem to fit our answer.
This is the assignment that was given to us for the Extension, so therefore, I will change the rules of the Hailstone Sequence.
The new problem I generated: if the number is even, divide it by 4 and add 1, and if the number is odd, multiply by 2 then add 2. These were the sequences I generated.
Here are some examples of how it turned out
16; 5, 12, 4, 10, 3.5
20; 6, 2.5
Unfortunately, I noticed in all cases that this sequence would eventually end in decimals - I tried many combinations, and none ever challenged that theory, similar to how the original sequence always ended in 4, 2, 1.
I really liked the extension of the problem though, as it challenged our thinking more than just explaining the problem - we had to mold the problem to fit our answer.