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A Rabbit ate the Carrots (Posted on 2005-02-05) Difficulty: 3 of 5
Find the common theme with all these numbers. ( Except for the fact that all of them are wrong )

242 + 32 = 450
252 + 567 = 1575
507 + 147 = 1200
1805 + 20 = 2205
570312 + 19208 = 798848

See The Solution Submitted by Syzygy    
Rating: 3.6667 (6 votes)

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Solution Puzzle Solution With Explanation | Comment 10 of 11 |
(In reply to answer by K Sengupta)

At the outset, we divide both sides of the first row by gcd of all
the three numbers in that row.

Now, in the first row, the three numbers are 242, 32 and 450, and gcd (242,32,450) = 2.

Then, the first row is now transformed to:
121 + 16 = 225
OR, 11^2 + 4^2 = 15^2

The above identity is obviously erroneous, but we note that inserting a square root ("v") symbol to each of the three numbers, we get the identity : 11 + 4 = 15, which is indeed true.

We are now in a position to conjecture that all the numbers in each of the five rows are missing the "v" symbol.

Accordingly, the requisite verification pertaining to all the the
five rows is furnished hereunder as follows:

First Row:

lhs = V242 + v32 = 11V2 + 4V2 = 15V2 = V450 = rhs

Second Row:

lhs = V252 + v567 = 6V7 + 9V7 = 15V7 = V1575 = rhs

Third Row:

lhs = V507 + v147 = 13V3 + 7V3 = 20V3 = V1200 = rhs

Fourth Row:

lhs = V1805 + v20 = 19V5 + 2V5 = 21V5 = V2205 = rhs

Fifth Row:

lhs = V570312 + v19208 = 534V2 + 98V2 = 632V2 = V798848 = rhs

Consequently, all the numbers in each of the five rows are
indeed missing the square root symbol.

Note:

Of course, the present exercise assumes that the "V" symbol
would yield only positive values. If the V symbol generated
positive values for some, but negative values for others, the
identity may not be valid.

Foer example, in the first row if V242 = 11V2 and V450 = 15V2,
but V32 = -4V2.

Then, lhs = (11 - 4)V2 = 7V2, which is not equal to rhs which is
15V2.


  Posted by K Sengupta on 2008-04-29 15:47:36
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