Number Sequences
On Uncertainty, Prediction Error, and Lost Reputation
Prediction Error and Loss of Reputation
Consider the following sequence:
1, 2, 3,…
Can you predict the next number in the sequence?
The correct answer is “6”.
If you, like most people, predicted “4”, you might be disappointed. You suffered a “prediction error”. You might have thought you had a “reputation for solving mathematical problems”, and that reputation just suffered a setback.
The next ten numbers in the sequence are:
6, 5, 5, 1, 6, 4, 2, 4, 1, 6, …
Uncertainty
If you think you can try and “solve” this sequence, don’t bother. Each number was generated randomly by picking a number between 1 and 6. It was purely chance (a one-in-216 chance, to be precise) that I got 1, 2, 3 at the beginning.
Before you get mad at me for challenging you to bogus puzzles, please consider the next game. You can reclaim your reputation.
Reputation reclaimed
Now, suppose we play the same game (where you try and guess the next number in the sequence). Suppose we continue to use the same “number generation strategy”, where I pick a number at random from 1, 2, 3, 4, 5, 6.
But suppose we add a new rule: You can choose to ignore specific numbers that you have chosen beforehand.
For example, if you choose to ignore 4, 5, and 6, our original sequence becomes:
1, 2, 3, 1, 2, 1
If you choose to ignore, 1, 4 , 5 and 6, it becomes:
2, 3, 2
You can choose to ignore all 6 numbers or none at all or anything in between.
Now, what is the optimal strategy to play this game? What set of numbers do you choose to ignore? What number do you guess?
Solution
Of course, the solution to this puzzle is simple and trivial. Your “ignore set” consists of any five numbers, say “1, 2, 3, 4 and 5” and you predict the remaining number, say “6”.
For example, our sequence is
6, 6, 6
And of course, you get 3 out of 3 predictions correct.
Philosophical Conclusion
Do you see the world for what it is? Or do you choose to see the world such that your reputation (for winning puzzles and life in general) is maximized?
If so, what is your ignore set, and what is your prediction?
