Maths Puzzle: Back to Black

I have a board that uses the numbers 1 to 90. I fill the board with Othello piece, they are black on one side and white on the other. All the pieces start black side up. First I flip every second number, so 2 becomes white, and 4, 6, 8 and so on. Then I flip every third number, then every fourth number, until finally I flip every 90th number. Which numbers end up black side up and why?

Music by Ghostly Dust Machine http://freemusicarchive.org/music/Ghostly_Dust_Machine/Bad_Panda_79/Ode_To_A_...

Category:

Education

Tags:

• maths
• math
• mathematics
• problem
• puzzle
• numbers
• factors
• divisors

[ashwinnarayanVlog]

Reminds me a lot of the Sieve of Eratosthenes which helps you find prime numbers.

Anyway, I'm a Numberphile/DeepSkyVideos/Sixt­ySymbols/periodicvideos subscriber and never realized you had your own awesome channel.

Hurrah \(◕ ‿ ◕)/ for mathematics!

[singingbanana]

@ashwinnarayanVlog I like to think of myself as the Green Ranger of numberphile. I got my own thing going on too.

[TyYann]

That's my 'Cells and Locks' math problem! Nice video, keep it up!

[singingbanana]

@TyYann Sorry mate! We're lucky not to double up more often. I only did it because I found this awesome prop! Still, I think 2 years is reasonable, we were so different then...

[Maibaum01]

Now, which piece was flipped the most times?

[livedandletdie]

@Maibaum01 the one with the most factors so it's 60, 72, 84, 90 which all of them have 12 factors.

[robbie4128]

@LoopyStudios wrote it in c++ it wasnt all that difficult but i was doing it at 2 am and my mistakes were just small little nit picky things that had to be fixed

[LoopyStudios]

@robbie4128 Doesn't seem that hard. What did you write it in.

[SFMacross]

The #'s listed in the examples are just the 'different' types of #'s that go into that particular #.

And you can see that they pair (taking away the last # [itself])..

First number times last = #

2nd # x 2nd to Last = #

3rd # x 3rd to last = #

Adding them up, you the 'flip count' for any given number. If you include 1 into the puzzle, you'll get the same results. Squares will be Odd, and non-squares will be even.

[SFMacross]

Meaning, if its flipped over an odd # of times it'll be white in the end.

Examples:

[32] - 2,4,8,16,32 [5*(Odd)]

[42] - 2,3,6,7,14,21,22 [7*(Odd)]

[90] - 2,3,5,6,9,10,15,18,30,45,90 [11*(Odd)]

However, the Perfect Square # is unique. Its the only number that has a number paired with itself, leaving only 1 number in that given set.

Examples:

[16] - 2,4,4,8,16 [4*(Even)]

[36] - 2,3,4,6,6,9,12,18,36 [8*(Even)]

All squares flip over an even amount of times leaving it black.

[SFMacross]

Before 4:23

I looked at this a great deal. Lots of proof using pictures.. And this is what I got.

If you choose any #, the only other #'s that effect it (turn it over), are the ones that go into it. That's obvious looking @ the vid, but its hard at first sight to know what that means.

When any given # goes into another, it is always paired with another different #. And, with its own # left over, it leaves an odd # of #'s going into itself.

[robbie4128]

@luigi90900 2,3,5,7,11,13,17,19,23,29,31,3­7,41,43,47,53,59,61,67,71,73,7­9,83,89

[robbie4128]

@Maibaum01 the 60th, 72nd, and 84th pieces were all flipped 11 times which is the most a single piece was flipped

[robbie4128]

i tired writing a program to do it............ it was tuff but i got it to do it :))