Brain Teasers
Integers of a Feather
In the finite set of integers 1 through 50000000, the following six integers share a characteristic that no other integers in the set share. What is the characteristic?
1, 36, 1225, 41616, 1413721, 48024900
1, 36, 1225, 41616, 1413721, 48024900
Answer
They are the only triangle numbers that are also squares of whole numbers.1 = 1^2
36 = 6^2
1225 = 35^2
41616 = 204^2
1413721 = 1189^2
48024900 = 6930^2
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Comments
Whats a triangle number?
A triangle number is the sum of integers from 1 thru x. 1 is a triangle number. 1 + 2 = 3, so 3 is a triangle number. 1 + 2 + 3 = 6, so 6 is a triangle number. 1 + 2 + 3 + 4 = 10, and so on.
This teaser was the hardest one I read so far
huh?
i don't get it!
i don't get it!
Jun 25, 2004
there is also another answer for this..
forgetting the special charaterstic of the integers.. one more feature is add the digits of each number u will get 1,9,10,18,19,27.. when u look at this the aletrnate numbers will have differenc of 9
forgetting the special charaterstic of the integers.. one more feature is add the digits of each number u will get 1,9,10,18,19,27.. when u look at this the aletrnate numbers will have differenc of 9
Going past the limit, the next number is 1631432881.
Each number is the square of the product of two numbers, each one of which is twice the preceding factor plus the one before that.
For example:
36 is the square of 6=2x3
1225 is the square of 35=5x7
41616 is the square of 204=12x17
and 12=2x5+2
17=2x7+3
Also if you subtract one of the given numbers from the next one, then square the difference, you get the number that is in the place that is the sum of the places of the two numbers in the subtraction:
For example:
(3rd minus 2nd number) 1225-36=1189 and
1189x1189=1413721 (5th number)
Each number is the square of the product of two numbers, each one of which is twice the preceding factor plus the one before that.
For example:
36 is the square of 6=2x3
1225 is the square of 35=5x7
41616 is the square of 204=12x17
and 12=2x5+2
17=2x7+3
Also if you subtract one of the given numbers from the next one, then square the difference, you get the number that is in the place that is the sum of the places of the two numbers in the subtraction:
For example:
(3rd minus 2nd number) 1225-36=1189 and
1189x1189=1413721 (5th number)
Blimey thats a bit hard isn't it?
me thinks this problems is very toughs ARRGGGHHH
(ADD A PIRATE'S ACCENT)
(ADD A PIRATE'S ACCENT)
lol, i don't get it... but nice job on it.
Ugh...this really shouldn't be the hardest teaser on the site. These are the sort of questions you find in IQ tests, these are easy peasy...
hard
hey sea anemone, do you have too much time on your hands? Sheesh! I got that they were perfect squares, and figure their prime factorization, but that was as far as i cared to go. Tough one.
Wow canu, honestly how in the WORLD did you come up with such connections? What goes on in your head that that pops up? That's absolutely astounding. Congratulations on having such a sharp mind.
Easy one if you already know it, probably not so easy when you don't.
In addition to the patterns given above, if a[n] is the nth term in the sequence then
a[n+1] = (a[n]-1)^2 / a[n-1]
for n > 2.
1225 = (36-1)^2 / 1
41616 = (1225 - 1)^2 / 36
1413721 = (41616 - 1)^2 / 1225
. . .
In addition to the patterns given above, if a[n] is the nth term in the sequence then
a[n+1] = (a[n]-1)^2 / a[n-1]
for n > 2.
1225 = (36-1)^2 / 1
41616 = (1225 - 1)^2 / 36
1413721 = (41616 - 1)^2 / 1225
. . .
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