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Tao takes us to the Collatz Conjecture. Proposed in 1937 by German mathematician Lothar Collatz, the Collatz Conjecture is fairly easy to describe, so here we go.
Take any natural number. There is a rule, or function, which we apply to that number, to get the next number. We then apply that rule iteratively, and see where it takes us. The rule is this: If the number is even, then divide it by 2; if the number is odd, then multiply by 3 and add 1. For example, consider 10. It’s even, so the rule says to divide by 2, taking us to 5. Now that’s odd, so we multiply 5 by 3 and then add 1, landing us on 16. Now 16 is even, so we cut it in half to get 8. Even again, so halving gets us 4. Now 4 is even, so we take half, getting 2, which is even, and cuts in half to 1.
Start with numbers other than 10, and you’ll still inevitably end at 1 … we think. That’s the Collatz Conjecture.
It’s definitely true for all numbers with less than 19 digits, so that covers whatever you probably had in mind. But even if computers check up to 100 or 1,000 digits, that’s far from a proof for all natural numbers.
The number of iterations it takes to first reach 1 is the “stopping time.” In the table below, we just call it number of "Stops".
Binary search tutorial: https://www.geeksforgeeks.org/binary-search-in-javascript/
function binarySearch(sortedArray, key) {
var start = 0;
var end = sortedArray.length - 1;
while (start <= end) {
var middle = Math.floor((start + end) / 2);
if (sortedArray[middle] === key) { // found the key
return middle;
}
else if (sortedArray[middle] < key) {
start = middle + 1; // continue searching to the right
}
else { end = middle - 1; // continue searching to the left
}
// key wasn't found
return -1;
}
| Argument | Stops | Explanation |
|---|---|---|
| 1 | 0 | 1 |
| 2 | 1 | 1 |
| 3 | 7 | 10 5 16 8 4 2 1 |
| 4 | 2 | 2 1 |
| 5 | 5 | 16 8 4 2 1 |
| 6 | 10 | 3 10 5 16 8 4 2 1 |
| 7 | 16 | 22 11 34 17 52 26 13 40 20 10 5 16 8 4 2 1 |
| 19 | 20 | 58 29 88 44 22 11 34 17 52 26 13 40 20 10 5 16 8 4 2 1 | 21 | 7 | 64 32 16 8 4 2 1 |
| 23 | 15 | 70 35 106 53 160 80 40 20 10 5 16 8 4 2 1 |