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Recognize rapid grid summation and trace the Collatz algorithm.
Using patterns rather than counting one by one.
When a grid contains repeated numbers, do we have to add them one by one? No! We can use grouping or multiplication to find the sum much faster. This is a great way to build your computational thinking.
Grid arrangements and sequence trees are necessary.
A clean scientific diagram split into two parts: on the left, a grid of squares with repeated numbers grouped by distinc…
Introduce Lothar Collatz and his 1937 puzzle.
Did you know that some math problems are so difficult that no one in the world has been able to solve them yet?
One of the most famous is the Collatz Conjecture. Proposed by the German mathematician Lothar Collatz in 1937, it is a deceptively simple number game that has baffled mathematicians for decades.
The even/odd rules of the sequence.
12 is even →
6 is even →
3 is odd →
Sequence continues: 10, 5, 16, 8, 4, 2, 1.
Trace the sequence starting at 12.
Generate the Collatz sequence starting with the number 12.
The rules are simple:
Complete the Collatz sequence for 17.
Let's trace the Collatz sequence starting at the number 17.
The rule states: if odd, multiply by 3 and add 1; if even, take half. Since 17 is odd, we multiply by 3 and add 1 to get .
This result is even, so we halve it to get 26, and halve it again to get 13. Because 13 is odd, applying the odd number rule gives us .
We then continue halving the even numbers: 20, then 10, then 5. Since 5 is odd, multiplying by 3 and adding 1 yields .
This even number then halves down through 8, 4, 2, and finally reaches 1.
Analyze why powers of 2 resolve quickly.
Problem: Recall the sequence of powers of 2, such as 2, 4, 8, 16, 32, ...
Why is the Collatz conjecture correct for all starting numbers in this sequence?
In the Collatz process:
Try following what happens to powers of 2.
Think about whether powers of 2 are odd or even.
Try examples like 32 → 16 or 16 → 8.
Start with any power of 2 and keep dividing by 2 until you reach 1.
The odd rule is 3n + 1. Think about whether we meet any odd number before reaching 1.
Write the complete idea in one or two sentences.