

An exercise in riddle-solving - techdog
http://asserttrue.blogspot.com/2009/05/exercise-in-riddle-solving.html

======
gjm11
Although the author says he solved it, he didn't; he made a guess and got
lucky. Perhaps there's a subtle joke going on? (Or maybe it's not so subtle
and I'm just being dim. Then again, I solved the puzzle in a couple of minutes
and he apparently didn't.)

I don't think it's a terribly interesting puzzle.

~~~
lhorie
His point is that he got the value of "?", but not the "correct" method to
arrive at that solution. Neither of blog posts seem to ask explicitly for the
method.

I think choosing to take a statistical gamble on a silly meaningless puzzle
(rather than attempt to solve it conventionally like everyone else) is
actually not necessarily a bad idea. I talked about that a while back here :
[http://lhorie.blogspot.com/2009/03/games-bias-and-
statistics...](http://lhorie.blogspot.com/2009/03/games-bias-and-
statistics.html) (albeit I was reacting to a more statistical-natured puzzle
at that time)

I do agree this puzzle isn't very interesting in that it doesn't evoke as much
gut feeling as, say, the Monty Hall paradox.

------
jpwagner
it jumps out right away that:

1+1+1+1+1 = 5

2+2+2+2+2 = 10 and 1+0 = 1

3+3+3+3+3 = 15 and 1+5 = 6

(whenever it's sums of digits you think mod 9)

since there are 5 numbers, the sum of the coefficients mod 9 = 5 (ie, 5, 14,
23...)

that's as far as I went before giving up

anyone got the rest without brute force?

\----------------------------------------------

Oh and what's with the author's justification of random guessing?

~~~
tspiteri
From

    
    
        1 1 1 1 1 = 5
        1 1 1 1 2 = 1
        1 1 1 1 3 = 6
        1 1 1 1 4 = 2
        ...
    

you can see that an increase of 1 in the last digit will increase the answer
by 5 or decrease it by 4, so it is an increase of 5 mod 9.

Similarly, from

    
    
        1 1 1 1 1 = 5
        2 1 1 1 1 = 8
        3 1 1 1 1 = 2
        4 1 1 1 1 = 5
        ...
    

you can see that an increase of 1 in the first digit will increase the answer
by 3 or decrease it by 6, so it is an increase of 3 mod 9.

So the first digit is multiplied by 3 and the last digit is multiplied by 5.
That rings a bell, since the last line is 3 1 4 1 5, which starts with a 3 and
ends with a 5, so you must suspect that the second digit should be multiplied
by 1, the third by 4, and the fourth by 1. Go over a few numbers, or all of
the numbers, to check this out, and it works, so you can apply it to the last
line and get the answer.

~~~
jpwagner
_since the last line is 31415_

That would be brute force,,,

~~~
tspiteri
There's a reason why the puzzle-writer used those 5 digits in the final
question, so that you can spot the pattern and find a shortcut. Spotting the
pattern is not brute force. To check your formula you have to do some
calculations, but that is something you cannot avoid. By going over the
numbers, what I mean is confirming that:

    
    
        1x3 + 2x1 + 3x4 + 4x1 + 5x5 = 1
        5x3 + 4x1 + 3x4 + 2x1 + 1x5 = 2
        ...
    

You can do that manually or mentally, so it is not what I call brute force.

And if you do not spot the 3 1 4 1 5 pattern, or just do not want to use the
shortcut provided, there is another way to do it. First let's find the second
digit. We only need our known digits and the two lines

    
    
        2 2 2 2 2 = 1
        1 1 2 2 2 = 6
    

Since we know that the multiplier of the first digit is 3, we can calculate
the value of 1 2 2 2 2 = 2 2 2 2 2 - 1 0 0 0 0 = 1 - 3 = 7. Then it follows
that the multiplier of the second digit is 1 2 2 2 2 - 1 1 2 2 2 = 7 - 6 = 1.
The third digit can be found by changing the line 1 2 3 4 5 = 1 to 3 3 3 4 4 =
3. You can do this since you have the multipliers for the first, second and
fifth digits. We already have the line 3 3 4 4 4 = 7, so the third digit is 7
- 3 = 4. For the fourth multiplier, change the line 1 1 1 1 1 = 5 to 2 2 2 1 2
= 0. Compare to 2 2 2 2 2 = 1 and you see that the fourth multiplier is 1. So
now you have all the five multipliers, 3, 1, 4, 1 and 5.

But we are not necessarily ready yet, we may need to add a constant, c. So
pick a line, the easiest seems to be 1 1 1 1 1 = 5. Now, since we know the
five multipliers, we can say that

    
    
        1x3 + 1x1 + 1x4 + 1x1 + 1x5 + c = 5
        5 + c = 5
        c = 0
    

The constant is 0. So just multiplying each digit by its respective multiplier
gives the correct answer. You still need to verify your formula by calculating
the right hand side of a few lines.

------
alexgartrell
my guess was pi :(

