10 million Accounts Puzzle, 1,000 followers problem

Algebra Level 5

Have you ever wondered how brilliant started? Well, here's the story of brilliant from its roots. Be like Sherlock Holmes and read carefully, there are clues to this problem scattered throughout the story, some are quite well hidden. I have bolded one part which may have been too hard to catch. Most of the info in this story is true except the parts about Daniel, Prasun, Brilliant's start date (I changed this for convenience) and the staff incident.

Assume today is May 11st. Exactly 2.5 years ago (Nov. 11th), Calvin, Sue, and Peter got together to form a site to attract the smartest kids in the world, each made his/her own account. Luring them not with rewards, but with intriguing challenges submitted by other users from around the world, Brilliant began to grow. The next day, Daniel, being the genius he was, became the first person to join this new site. He began making thousands of the best problems the all knowing Gods of this universe had ever seen.

Unfortunately, that same day, Prasun, while trolling people on Omeggle , saw an ad for a math website by the title "brilliant.org". He moved most of his trolling focus from Omeggle to Brilliant and the growth rate drastically slowed down.

Also on that very day, Calvin, being the math God he is, correctly hypothesized that Brilliant's growth rate would take the form # of accounts = a 5 b x + c \text{\# of accounts}=\lfloor a\cdot 5^{bx}+c\rfloor where a , b , c a,b,c are constants, a , c a,c are integers, and x x is the amount of time (measured in months) elapsed from Nov. 11th. Unfortunately, he had to change it when Prasun the troll joined to the new growth rate # of accounts = a e b x + c \text{\# of accounts}=\lfloor a\cdot e^{bx}+c\rfloor where e e is the natural logarithm base.

Many moons passed and brilliant continued to grow at the exponential rate law mentioned immediately above.

One groggy Thursday morning while Peter was dozing off at his desk, Silas burst into the room screaming at the top of his lungs, "We did it! We hit 1 million accounts!". Peter, alarmed, immediately woke up to see a jubilant Silas on the other side of his work cubical. Silas began to run around the room and Aaron soon joined him. David walked into the room after hearing all the commotion only to see a smiling Silas and equally joyful Aaron barreling down the isle straight towards him. BAM! The three collided in what can only be described as "the second Big Bang". After the stars disappeared from the three Stooges' eyes, David realized his phone was broken. Being the physics genius he is, he immediately began to calculate how much force was required to break his phone's 1/32 inch thick anodized aluminum backplate. At the same time, Silas and Aaron promised to buy David a new phone since Christmas was coming up in 14 days.

As days turned to weeks and weeks to months, Brilliant became swamped with even more new members. Sue realized that Brilliant needed more staff members so she posted this note looking for more people to hire. In this note, she stated that Brilliant had 1.4 million accounts. However, this was actually a slight miscount due to technical errors. Today, May 11th, 2015, Brilliant finally reached exactly 1.4 million accounts.

Based on the information given above, in which month and year will Brilliant reach 10 million accounts?


Details and Assumptions :

  • For simplicity's sake, assume all months have 30 days.

  • The floor function is just for "technical correctness" (you can't have 10.5 accounts). You don't need it in your calculations as it won't affect the result (assuming it's correct) by more than a day. And even being 7 days off won't affect your answer.

  • You may use a computer for equation solving. When working with b b , I suggest using at the least the ten-thousandths place,.

  • 1 million is the number one followed by 6 zeros.

Note that in reality, my prediction is that it might take a little longer to reach 10 million accounts than the actual answer to this question suggests.

This is my unofficial 1000 followers problem. I don't know how much attention it will get because it's a new type of problem. Tell me how you like it and I might make another one. So please, if I may ask, do take the time to at least read it.
February 2018 June 2019 December 2018 June 2018 April 2018 February 2019 April 2019 December 2017

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1 solution

Trevor Arashiro
May 11, 2015

Clues explained:

" 2.5 years ago (Nov. 11th), Calvin, Sue, and Peter got together..." \text{" 2.5 years ago (Nov. 11th), Calvin, Sue, and Peter got together..."}

Brilliant started 30 months ago and had 3 accounts at time x = 0 x=0 . Therefore, we have our equation to be

a 5 0 b + c = 3 a and c are integers, so the floor doesn’t matter \lfloor a\cdot 5^{0\cdot b}+c\rfloor =3~ \text{a and c are integers, so the floor doesn't matter}

c = 3 a \Longrightarrow c=3-a (eq. (i))

Had a and d not been integers, the floor actually would have been crucial for this step as it would have allowed for (a,b) to vary quite a bit. This is the only step the floor function would have been necessary for calculations had a and b not

"Again correctly hypothesized that \text{"Again correctly hypothesized that} # of accounts = a e b x + c \text{\# of accounts}=\lfloor a\cdot e^{bx}+c\rfloor ".

Here we are simply changing the growth rate.

"One groggy Thursday morning while Peter was dozing off.....since Christmas was coming up in 14 days ."

This was the trickiest part. I wasn't sure if it was too hard, so I bolded the important terms. The 14 days before Christmas means that this is December 11th. The fact that Christmas is happening on a Thursday means that this was the December of LAST YEAR. Why might you ask? Because it's the only Christmas within the past 3 years that's happened on a Thursday.

"’WE DID IT, WE HIT 1,000,000 ACCOUNTS’." \text{"'WE DID IT, WE HIT 1,000,000 ACCOUNTS'."}

This is our second data point. It occurs at time x = 25 x=25 (25 months after Nov. 11th) thus we have the equation

1 , 000 , 000 = a e 25 b + c 1,000,000=a*e^{25\cdot b}+c substituting for c c from (eq. (i)).

1 , 000 , 000 = a e 25 b + 3 a 1,000,000=a*e^{25\cdot b}+3-a

"May 11th, 2015, Brilliant finally reached exactly 1.4 million accounts." \text{"May 11th, 2015, Brilliant finally reached exactly 1.4 million accounts."}

This one is pretty self explanatory. It's simply the third data point.

1 , 400 , 000 = a e 30 b + 3 a 1,400,000=a*e^{30\cdot b}+3-a


From all of this, we have the three equations

{ 1 , 400 , 000 = a e 30 b + 3 a 1 , 000 , 000 = a e 25 b + 3 a c = 3 a \begin{cases} 1,400,000=a*e^{30\cdot b}+3-a \\ 1,000,000=a*e^{25\cdot b}+3-a \\ c=3-a \\ \end{cases}

Solving this system, we get our equation to be 396191 e ( x 0.0503842 ) 396188 = 396191*e^(x*0.0503842)-396188= # of followers.

Thus we will reach 10,000,000 at

396191 e ( x 0.0503842 ) 396188 = 10 , 000 , 000 x = 64.8477 396191*e^(x*0.0503842)-396188=10,000,000\Longrightarrow x=64.8477

Thus the correct date will be 5 years, 4 months, 26.27 days from 2.5 years ago . Or Apr. 7th 2018 \boxed{\text{Apr. 7th 2018}}

The most attractive problem I've seen on brilliant!! +10

Aditya Kumar - 6 years, 1 month ago

Here's a challenge for everyone, comment below when you think brilliant will hit 10,000,000 accounts. And in 3 or so years we''ll find out! My guess is Nov. 10th 2018.

Trevor Arashiro - 6 years, 1 month ago

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I'm guessing that the growth rate will level out somewhat, (although remain impressive), and that the number of accounts will reach 10,000,000 mid-2022. I have often wondered, though, if the number Sue gives refers to the number of Facebook "likes" or the number of actual accounts on this site. Also, I understand that there are also some people who have multiple accounts, accessed through the various portals available. We used to be able to see the number of "followers" of each subject but that doesn't seem to be the case anymore; the last time I remember seeing those values there were roughly 450,000 members following Algebra, (which I would surmise is the most popular subject to follow), and thus thought that there might be perhaps half a million actual members.

Regardless, the site is a "happening thing" and the best is yet to come. :)

Brian Charlesworth - 6 years, 1 month ago

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Congrats on 1000 followers! Looks like you weren't too far off the mark at all.

When will you post your 1000 followers problem? I'm dying to solve it as I'm sure you have something great in store :)

Trevor Arashiro - 6 years ago

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@Trevor Arashiro Thanks! I do want to come up with a special question, but thus far the ideas I've had have evaded solution. You've set the bar high with this question; I'm surprised it wasn't put on the home page, but it had nevertheless proved to be extremely popular. :)

Brian Charlesworth - 6 years ago

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@Brian Charlesworth Yes, I'm quite happy with how popular this got, guess the 2-3 hours of work it took to make this weren't for nothing.

Pretty much any of your previous questions could qualify as an honorary milestone problem because they're all just that good :)

P.S. you should make a note detailing all the "un-answerable" questions you've thought of. I'm sure they're all disgustingly hard if not impossible, but I and many others would love to read them as some might inspire a problem or two to be created.

Trevor Arashiro - 6 years ago

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@Trevor Arashiro Thanks for the encouragement, :) I just need to stop getting distracted and get a good problem posted soon. And that might be a good idea; post a note with 5 or so problems that I have dreamt up but couldn't resolve and see what kind of response it gets. I have done this with a single problem before which went nowhere, but I have more followers now so perhaps it would gain better traction this time around.

Brian Charlesworth - 6 years ago

@Trevor Arashiro Congratulations on having 1000 followers! I am your 1000 follower as I unfollowed and followed you again :P ;)

Raghav Vaidyanathan - 6 years, 1 month ago

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I have used this technique quite a lot !

For example , with Satvik Golechcha , Krishna , and some other guys :P

A Former Brilliant Member - 6 years, 1 month ago

A really nice problem Trevor! (It took me over an 20 mins to solve this :/ ).Upvoted.

Venkata Karthik Bandaru - 6 years, 1 month ago

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Thanks! Glad you liked it. Lol, it took like 3 hours to make.

Trevor Arashiro - 6 years, 1 month ago

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You are really perseverant Trevor. And obviously, it takes a lot more time to create a problem than to solve it. I like these Detective kind of problems from childhood and do keep posting problems like this. I hope you won't mind the use of WolframAlpha for those ugly calculations.

Venkata Karthik Bandaru - 6 years, 1 month ago

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@Venkata Karthik Bandaru Oh, of course, I'd love to post more of these.

Btw, you'd have to be insane NOT to use wolfram, which is why I stated it in the details and assumptions.

Trevor Arashiro - 6 years, 1 month ago

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@Trevor Arashiro Oh lol, I didn't notice it!

Venkata Karthik Bandaru - 6 years, 1 month ago

Contrary to popular belief, I was not the first person to join Brilliant XD

I know that a lot of people were already on brilliant by the time I joined (It was during the time when Calvin made 8 problems per week, and I remember seeing hundreds of people completing all 8 problems when I couldn't do so)

Also sorry for not making problems recently :(

Daniel Liu - 6 years, 1 month ago

Congratulations Trevor !

A Former Brilliant Member - 6 years, 1 month ago

@Trevor Arashiro It is 16th May, 2018 now.......What is the status of the accounts now??

Aaghaz Mahajan - 3 years ago

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