Reminds me of a problem I was once set at school. What comes next: 1, 8, 27, 64...
I gave 117. The difference between consecutive numbers is 7, 19, 37: the 4th, 8th and 12th prime numbers. So, of course, the next difference is the 16th prime number: 53.
Perfectly logical, and completely missing the obvious solution!
- Joined
- 10 Dec 06
- Moves
- 8528
Originally posted by mtthwThe pattern is also cubes of the natural numbers, how can one solution be chosen over the other.... ?
Reminds me of a problem I was once set at school. What comes next: 1, 8, 27, 64...
I gave 117. The difference between consecutive numbers is 7, 19, 37: the 4th, 8th and 12th prime numbers. So, of course, the next difference is the 16th prime number: 53.
Perfectly logical, and completely missing the obvious solution!
Edit: nevermind you were wrong..lol🙂😵
Edit again: that math teacher should have given you credit for looking to deeply
So what is the next number of a serie of numbers?
Any number is correct if you do have a well motivated method for chosing that number.
This means that the authors choice of next number is only one of many solutions.
What is the next number after 7, 5, 5, 4 ?
The correct solution is 7, but why?
And what is the next number after that?
I can imagine that you can come up with a very plausible solution, without even being near to mine. The number is not interesting, the method is.
Originally posted by mtthwThere's a similar pattern with the squares of numbers:
Reminds me of a problem I was once set at school. What comes next: 1, 8, 27, 64...
I gave 117. The difference between consecutive numbers is 7, 19, 37: the 4th, 8th and 12th prime numbers. So, of course, the next difference is the 16th prime number: 53.
Perfectly logical, and completely missing the obvious solution!
1, 4, 9, 16, ... have differences of
1, 3, 5, 7, ...
Originally posted by forkedknightThe difference is that yours was the right solution - the difference between consecutive squares are consecutive odd numbers (easy enough to prove). In my case I came up with a completely different sequence that happened to coincide for the first few numbers.
right, they are all consecutive odd numbers
My point was I did something similar to mattw in school given the sequence of squares.
Determining the next number in a sequence where the rule is not given is an inductive exercise.
What this means is that you have to determine the rule from the given facts, rather than determining the facts from a given set of rules (deductive problems).
Inductive reasoning problems have a curious property that they can never be proven absolutely, for any set of facts, there are an infinite number of diverging rules which can account for them. The reason many 'next-in-sequence' problems are deemed acceptable is that for the given sequence, there may be one particular rule is particularly elegant, simpler, and less arbitrary than any other potential rule, and which most people would agree is the best solution.
1, 4, 9, 16, 25,.. is recognized as the sequence of squares for example.
1, 1, 2, 3, 5, 8, 13, .. is another pattern that has one particularly simple rule.
For the record, problems of the 'which doesn't belong' sort suffer even more heavily from the inductive nature of the problem type.