## Monday, April 16, 2012

### Counting arguments fail, think better!

Here is a puzzle (the answer is below, do not scroll unless you want to see it!):

8809=6            5555=0
7111=0            8193=3
2172=0            8096=5
6666=4            1012=1
1111=0            7777=0
3213=0            9999=4
7662=2            7756=1
9313=1            6855=3
0000=4            9881=5
2222=0            5531=0
3333=0            2581=?

This one was posted in Lifehacker, the author took it from a flying Facebook post.
The clue provided was that the kids could get the answer faster than the educated thinkers who tend to think in more complex ways.  I can  imagine if your brain is going round the world to solve the problem.
I would urge you to try and guess the answer, which is much fun! Please still do not scroll.
I will even put a youtube video of Simon's cat just to avoid the answer to be seen.

:)

:)

2581=2.

The answer is arrived once you realize that there are two "o" in 8. Go back and check the other answers in the list counting the "o"s in the numbers on the left hand side of the puzzle.

However, this counting argument fails miserably when the numbers are written in another script.
Here, the same example is being provided in coded form of Devanagari numbers. Suppose the left hand side is the code, and right hand is the answer to the code):
८८०९ =6            5555=0
७१११ =0            8193=3
२१७२ =0            ८०९६ =5
६६६६ =4            १०१२ =1
११११ =0            ७७७७ =0
३२१३ =0            ९९९९ =4
७६६२ =2            ७७५६ =1
९३१३ =1            ६८५५ =3
०००० =4            ९८८१ =5
२२२२ =0            ५५३१ =0
३३३३ =0            २५८१ =?
In this case, You can not count the "o" on the left hand side to arrive at the (same) answer!

>
--> ---
Symbols: compute the values of the symbols.
Well,  think of the numbers as "variable" So that
०  ०  ०  ०  = 4 means: "०"+"० "+"० "+"० " = ४
ie 4 of "० " = 4
=> "० " = 1
Similarly,
६६६६  = 4
=> "६ " = 1.
9999=4
=> "९ " = 1
and, २२२२ =0 gave: "२ " = 0
११११ =0 => "१ " = 0
३३३३  = 0 => "३ " = 0
५५५५  = 0 => "५ " = 0
७७७७  = 0 => "७ " = 0
Now do a test case: २१७२  = 0
LHS= "२ "+"१ "+"७ "+"२ " = 0+0+0+0 = 0 = RHS
proved!
Now, to solve, ८८०९  = 6
Suppose, " "=y
Substitute values
y+y+1+1 = 6
2y+2 = 6,
=> 2y = 4
=>y=2
Therefore, y="८ "=2

We tested it for all the examples.
Therefore, we can say:
2581 = "२ "+"५ "+"८ "+"१ " after substitution:
= 0+0+2+0 =2

The answer is 2. The method is more general

The addition is still the counting argument isn't it?
:P

(Thanks to my wife for bringing this solution to my attention.)