Brain Teasers
The Romankachi Siblings !!!
We were all celebrating the 25th birthday of my beloved sister, Lavi the kindhearted Romankachi. She was younger to me Rudy the business-minded Romankachi by two years and to our elder brother Fred the well-settled Romankachi by four years. Just as the celebrations were in full swing, I realized something about our ages. I found that I could say at least two things definitely about each of the following two integer values formed by manipulating our ages taken as integers:-
a) The number formed by adding 1 to the product of Lavi's and my age considered during any one year right since she was a year old.
b) The number formed by adding 1 to the product of Fred's and my ages considered during any one year right since I was a year old.
Though I forgot about this the next moment and got busy enjoying myself, could you tell what were the two things that flashed across my mind.
Consider all ages to be integer values. Also during any given year, Lavi was exactly two years younger to me and exactly four years younger to Fred.
a) The number formed by adding 1 to the product of Lavi's and my age considered during any one year right since she was a year old.
b) The number formed by adding 1 to the product of Fred's and my ages considered during any one year right since I was a year old.
Though I forgot about this the next moment and got busy enjoying myself, could you tell what were the two things that flashed across my mind.
Consider all ages to be integer values. Also during any given year, Lavi was exactly two years younger to me and exactly four years younger to Fred.
Hint
I'm exactly two years elder to Lavi and exactly two years younger to Fred.Answer
a) The number formed by adding 1 to the product of Lavi's and my age considered during any one year right since she was a year old.The two things that I could definitely say about the above integer value are :-
1) It is a perfect square.
2) The number formed is the square of a number 1 less than my age in that year.
Example: When Lavi was 3, I was 5 -> (3*5)+1 gives 16 which is the square of 4, which is 1 less than my age(5) in that year.
b) The number formed by adding 1 to the product of Fred's and my ages considered during any one year right since I was a year old.
The two things that I could definitely say about the above integer value are :-
1) It is a perfect square.
2) The number formed is square of a number 1 more than my age in that year.
Example: when I was 6, Fred was 8-> (6*8)+1 gives 49 which is the square of 7, a number 1 more than my age(6) in that year.
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An interesting scenario but a very vague question. I would like to see a worked solution showing how the answer was arrived at. (Even so I think this teaser deserves better than its current rating.)
cool
After a little effort to understand what the teaser was asking, I knew the answer immediately.
There are many ways to show why this is true, but the simple explanation is that
(x+1) * (x - 1) = x^2 - 1
in fact, that can be generalized further to:
(x + y) * (x - y) = x^2 - y^2
This yeilds all sorts of interesting and useful identities, such as the difference between the squares of X and X + 1 is X + X + 1. This is useful for quickly calculating squares of numbers ending in 1 or 9.
31^2 = 30^2 + 30 + 31 = 900 + 30 + 31 = 961
49^2 = 50^2 - 50 - 49 = 2500 - 50 - 49 = 2401
Another interesting thing that could have occurred to Rudy is that multiplying his sister's age times his brother's age and adding four yeilds the square of Rudy's age.
Or Rudy might have observed that multiplying his sister's age times his brother's age times the square of Rudy's age and adding four yeilds a perfect square.
Or perhaps Rudy is a real pinhead and observed that if he takes the fourth power of his age and subtracts the product of the squares of the ages of his brother and sister, then divides this difference by 8 and subtracts 2 he gets the product of the ages of his brother and sister.
Fun, fun, fun!
There are many ways to show why this is true, but the simple explanation is that
(x+1) * (x - 1) = x^2 - 1
in fact, that can be generalized further to:
(x + y) * (x - y) = x^2 - y^2
This yeilds all sorts of interesting and useful identities, such as the difference between the squares of X and X + 1 is X + X + 1. This is useful for quickly calculating squares of numbers ending in 1 or 9.
31^2 = 30^2 + 30 + 31 = 900 + 30 + 31 = 961
49^2 = 50^2 - 50 - 49 = 2500 - 50 - 49 = 2401
Another interesting thing that could have occurred to Rudy is that multiplying his sister's age times his brother's age and adding four yeilds the square of Rudy's age.
Or Rudy might have observed that multiplying his sister's age times his brother's age times the square of Rudy's age and adding four yeilds a perfect square.
Or perhaps Rudy is a real pinhead and observed that if he takes the fourth power of his age and subtracts the product of the squares of the ages of his brother and sister, then divides this difference by 8 and subtracts 2 he gets the product of the ages of his brother and sister.
Fun, fun, fun!
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