Hopefully interesting challenge :P

[Kevin]

http://www.truploader.com/view/623425

Open the link and you will be brought to the logic challenge. This is the same as what I have posted before as a written challenge, but this looks much better because it's animated and you won't have to look at tons of things immediately, but see them one by one.

If you can find the answer without opening the hint then you're awesome!
If you use some hints, please tell me how many hints you needed (there are #1 to #5, try to find the answer using fewest hints, but I will guess most people will need to get at least to #4 before finding the answer)

Don't write the answer though, we don't want to spoil it to others.
If you already read the answer earlier, you can still share this to friends .

[Reaper]

Nice challenge there. I did figure out a method that works without using any hints, though I'm not sure how to put it into three words properly. Alright, got one.

Edit: There's apparently a bunch of possible solutions, all of which will yield different results if you use digits other than 3,6 and 9. I've got three "normal" ones and about 5 which are either crazy or just rather close to one of the normals. I guess you can find a lot more of crazy(and perhaps also normal) ones though. :P

[Kevin]

^
Yeah, I forgot to mention that reactions only works for 0, 3, 6 and 9 starting circles in each boxes, and actually having some empty boxes in the middle isn't a problem :P . Wow, I don't think I will be able to solve it without seeing the hints if I were not the creator xD .

!! (open only if you have finished the challenge because it spoils the answer ^^ )

Spoiler (Click to View)

Actually I found the pattern accidentally when playing with my calculator during some boring or too easy studies at my school. Firstly, I found out that
3x4 = 12
33x4 = 132
333x4 = 1332
3333x4 = 13332
Later I found out it works for 6 and 9 too,
66x4 = 264
666x4 = 2664
6666x4 = 26664

It doesn't work with any other number. It's funny because it seems like the last digit gives 1/3 of it's value to make a new digit on the left.
I made my first quiz and most of my friends can figure it out:
f(3) = 12
f(33) = 132
f(333) = 1332
f(3333) = 13332
What the heck is f(x)?

Soon I noticed that it happens because 3x4 = 12, so
33333x4 = 120000 + 12000 + 1200 + 120 + 12, the number one carries through all the way, converting all number '2' back into 3.

Then somehow I got the idea to combine these numbers.
36x4 = 144
336x4 = 1344
3336x4 = 13344
First, I thought it fails to work, as well as when using other numbers. Soon, though, I realized that each digit just give 1/3 of its value to the previous digit.
The first funny ruleset that I found is very interesting for me. There is a bag, which walks from the right to the left.
It always wants to carry 1/3 of the value of the digit it meets.

For example, 3669693 x 4 = ...?
The bag first comes to the last digit, i.e. number 3, so it takes 1 of it. So the answer's last digit must be 2 (because 1 is taken away by the bag).
Next, when it meets the number 9, the bag wants to have 3. Since it already have 1, it only take 2, thus the number 9 decreases to 7.
When it meets the next number, 6, the bag only want to have 2 while it already have 3, so it gives 1 to the number 6 and it becomes 7.

This works everytime, if the number consists of only the numbers 0, 3, 6 and 9.

I guess, when I reverse it, people may realize the more complex patterns, but not that it is actually as simple as a multiplication of four.

[zort]

!! (open only if you have finished the challenge because it spoils the answer ^^ )

Heh, [ spoiler]s aren't actually spoilers anymore. (The following one is though.)

Spoiler (Click to View)

If you know modular arithmetic (seems like you don't?), you'll note that 4 ≡ 2 * 3^-1 mod 10. 0, 3, 6, and 9 are the only numbers modulo 10 that are divisible by 3, so they're the only ones for whom dividing by 3 modulo 10 is the same as dividing by 3 not-modulo-10. Long story short, here are the other numbers besides 4 with this property:

1 * 1^-1 ≡ 1 mod 10 and (1 * 1^-1 mod 10) = 10 - 9*1/1, so e.g. 1*1111 = 01111
1 * 3^-1 ≡ 7 mod 10 and (1 * 3^-1 mod 10) = 10 - 9*1/3, so e.g. 7*3333 = 23331
2 * 3^-1 ≡ 4 mod 10 and (2 * 3^-1 mod 10) = 10 - 9*2/3, so e.g. 4*6666 = 26664
3 * 3^-1 ≡ 1 mod 10 and (3 * 3^-1 mod 10) = 10 - 9*3/3, so e.g. 1*6666 = 06666
5 * 7^-1 ≡ 5 mod 10 and (5 * 7^-1 mod 10) = 10 - 9*5/7, so e.g. 5*7777 = 38885 <- uhh, not sure what's going on here
1 * 9^-1 ≡ 9 mod 10 and (1 * 9^-1 mod 10) = 10 - 9*1/9, so e.g. 9*9999 = 89991
2 * 9^-1 ≡ 8 mod 10 and (2 * 9^-1 mod 10) = 10 - 9*2/9, so e.g. 8*9999 = 79992
3 * 9^-1 ≡ 7 mod 10 and (3 * 9^-1 mod 10) = 10 - 9*3/9, so e.g. 7*9999 = 69993
4 * 9^-1 ≡ 6 mod 10 and (4 * 9^-1 mod 10) = 10 - 9*4/9, so e.g. 6*9999 = 59994
5 * 9^-1 ≡ 5 mod 10 and (5 * 9^-1 mod 10) = 10 - 9*5/9, so e.g. 5*9999 = 49995
6 * 9^-1 ≡ 4 mod 10 and (6 * 9^-1 mod 10) = 10 - 9*6/9, so e.g. 4*9999 = 39996
7 * 9^-1 ≡ 3 mod 10 and (7 * 9^-1 mod 10) = 10 - 9*7/9, so e.g. 3*9999 = 29997
8 * 9^-1 ≡ 2 mod 10 and (8 * 9^-1 mod 10) = 10 - 9*8/9, so e.g. 2*9999 = 19998
9 * 9^-1 ≡ 1 mod 10 and (9 * 9^-1 mod 10) = 10 - 9*9/9, so e.g. 1*9999 = 09999

Hmm, on second thought I don't know what's going on.

How long did it take you to make the flash thing btw? It's pretty professional.

[Kevin]

Hmm, I didn't know that . I understand modulus in programming :P (which results in the remainder of the division). I know the congruent sign, but I don't really know why you don't use the equal sign instead, and what those ^-1 means, so in the end I don't understand what you were explaining, even after you writing all those examples xP . But I do notice these special numbers that do similar things when multiplied with any 0-3-6-9 numbers :P , and the relation with mod 10 seems very logical.

At first glance I thought the ^-1 is the power, but doesn't seem appropriate there :P . Next I think about multiplying with the number -1, but that's not the usual way to write and when I reach 1x9[sup]-1[/sup ≡ 1 mod 10, it doesn't seem to be multiplication by -1 anymore XP .

My school doesn't teach modulus xD or modulo O_o I'm not familiar with modulo, but it seems they are close or the same.