Triangle Numbers
Series teasers are where you try to complete the sequence of a series of letters, numbers or objects.
Triangle numbers are formed by adding successive integers, i.e.
1 = 1+0,
3 = 1+2,
6 = 1+2+3,
10 = 1+2+3+4,
etc.
So the first few triangle numbers would be:
1,3,6,10,15,21,28, ...
Suppose somebody tells you a random integer. How can you quickly find out whether or not it is a triangle number?
HintThe method is linked to another common series.
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Answer
Multiply the number by 8 then add 1. If the result is a perfect square, then the original number is a triangle number.
For example: 10*8+1 = 81. 81 = 9^2. So 10 is a triangle number.
11*8+1 = 89. 89 is not a perfect square, so 11 is not a triangle number.
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Comments
jimbo
Sep 19, 2003
 Cool teaser! Here is another way. Double it. find the square root (approx) and round up and down to consecutive integers. If they give the double, it is triangular. Example 15 doubled is 30. Since it lies between 25 and 36 is square root is about 5.5. Rounding up and down make 5 and 6. 5x6 = 30 Yes! 
smithy
Sep 19, 2003
 Nice method jimbo!  but I think my one's a little less complicated... 
Mr_R
Dec 13, 2003
 I'm scratching my head as to why the given solution works? Good mind itch though. I had to look at the answer. 
(user deleted)
Jan 18, 2005
 I don't get it! 
(user deleted)
Jan 18, 2005
 I don't get it! 
brianz
May 12, 2005
 Wait...
A triangular number is of the form (n^2+n)/2 where n is an integer.
4n^2+4n+1 is that times 8 and then plus one which can be factored out to be (2n+1)^2
The square root of that is 2n+1 and is an integer as long as n is an integer or half an integer. Now if n is something like 1.5, (n^2+n)/2 will not be an integer. Therefore, your method works.
Good one! 
lessthanjake789
Jun 02, 2005
 is there any particular mathematical reasont hat works out, or is it just a rnadom quirk of mathematics that applies?
(also, once into bigger numbers, that method wont be so quick, trying to figure out perfect square and what not, but this is interestingly related to partial sum formula, because thats what triangle numbers are, sum of all integers from 1 to X. so for any number x, Sum = (x/2)*(2+(x1)) for example, x=10, sum = 5*(2+9) = 55, t'works. dont bother trying to figure out the summation formula, I am telling you it works 
fatjohn
Aug 05, 2005
 Never heard it explained with the multiply it by 8, add 1 and take its square solution. Nice twist, and good teaser. 
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