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At some point I gave the Collatz conjecture some thought and understood the probabilistic intuition of why it should be the case. Although I obviously couldn't

At some point I gave the Collatz conjecture some thought and understood the probabilistic intuition of why it should be the case. Although I obviously couldn't prove it, I decided to write my thoughts on why it is intuitively obvious [1]. A few months later I received a comment mentioning that a proof has been submitted based on the same intuition [2]. [1] http://nocurve.com/virtual-lab/contemplating-the-collatz-conjecture-3n1/ http://nocurve.com/virtual-lab/contemplating-the-collatz-con... [2] http://nocurve.com/about/ http://nocurve.com/about/

Liron 6y ago
> Every time the coin turns out heads, I take half of the money on the table (i.e the total you have left is divided by 2). Every time it turns out to be tails, your money is multiplied by 1.5 (plus a 50 cents bonus). It’s obvious that this game is totally rigged to my advantage as the amount of heads will converge to be similar to the amount of tails, but the loss you incur with each head result (money divided by 2) is larger than the win gained with each tail result (money multiplied by 1.5 + change). This game has positive expected value, the only problem is risk of ruin if you size your bets too aggressively as a percentage of your chip stack
- beojan 6y ago
  You can't be ruined if you can only lose a percentage of what you bet.
niknoble 6y ago
This type of reasoning only gives you intuition about half the conjecture. The conjecture says two things: 1. No sequence grows bigger and bigger forever; and 2. No sequence gets caught in a loop (except for the trivial 4 -> 2 -> 1 -> 4 loop). Your article addresses 1, but I don't think it says much about 2. Sure, sequences should probably trend downward over time, but why shouldn't a sequence that hops upward at the beginning get snagged on a value it's already hit as it falls back down, trapping it a nontrivial loop?
- placebo 6y ago
  Both parts you mentioned are derived from the Wikipedia formulation which is that "no matter what value of n, the sequence will always reach 1", but the subject of possible non trivial loops is an interesting one to contemplate from the coin toss perspective. The equivalence I made is to a truly random coin toss where heads means the number will be divided by two, and tails means the number will be multiplied by 1.5 and then 0.5 added to the result. If the coin toss is truly random, it stands to reason there will be no loop (following the same intuition that entropy always increases). Any permanent loop will contradict the randomness of the coin toss. Whether or not the reasoning above gives valid intuition to the conjecture being true depends upon the degree to which the Collatz mathematical procedure can be treated as generating a truly random process, and this is the point where I presented the main flaw in the intuition.

Liron 6y ago

> Every time the coin turns out heads, I take half of the money on the table (i.e the total you have left is divided by 2). Every time it turns out to be tails, your money is multiplied by 1.5 (plus a 50 cents bonus). It’s obvious that this game is totally rigged to my advantage as the amount of heads will converge to be similar to the amount of tails, but the loss you incur with each head result (money divided by 2) is larger than the win gained with each tail result (money multiplied by 1.5 + change). This game has positive expected value, the only problem is risk of ruin if you size your bets too aggressively as a percentage of your chip stack

beojan 6y ago
You can't be ruined if you can only lose a percentage of what you bet.

niknoble 6y ago

This type of reasoning only gives you intuition about half the conjecture. The conjecture says two things: 1. No sequence grows bigger and bigger forever; and 2. No sequence gets caught in a loop (except for the trivial 4 -> 2 -> 1 -> 4 loop). Your article addresses 1, but I don't think it says much about 2. Sure, sequences should probably trend downward over time, but why shouldn't a sequence that hops upward at the beginning get snagged on a value it's already hit as it falls back down, trapping it a nontrivial loop?

placebo 6y ago
Both parts you mentioned are derived from the Wikipedia formulation which is that "no matter what value of n, the sequence will always reach 1", but the subject of possible non trivial loops is an interesting one to contemplate from the coin toss perspective. The equivalence I made is to a truly random coin toss where heads means the number will be divided by two, and tails means the number will be multiplied by 1.5 and then 0.5 added to the result. If the coin toss is truly random, it stands to reason there will be no loop (following the same intuition that entropy always increases). Any permanent loop will contradict the randomness of the coin toss. Whether or not the reasoning above gives valid intuition to the conjecture being true depends upon the degree to which the Collatz mathematical procedure can be treated as generating a truly random process, and this is the point where I presented the main flaw in the intuition.