On my first day, at my first lecture at university, my math’s professor wrote ** { the second 10-digit prime in consecutive digits of e }.com **on the blackboard. He then proceeded to give an hour long lecture, without once referring to it. At the end, someone asked what it was. His response was simply “that’s for you to find out”. He then left.

A week later it was still there, and still no one knew what it was. We decided that it had to be something to do with Euler’s constant (e = 2.71828….), and thus decided that it was simply (haha) asking for the second 10 digit prime in the decimal expansion of *e*. Initially we wrote out *e* to about 40 decimal places (e = Sum_{k >= 0} 1/k! = lim_{x -> 0} (1+x)^(1/x) ), but struggled to find what we were looking for:

2.**7182818284**590452353602874713526624977572 – not a prime

2.7**1828182845**90452353602874713526624977572 – not a prime

2.71**8281828459**0452353602874713526624977572 – not a prime

2.718**2818284590**452353602874713526624977572 – not a prime

And so on.

In a fit of frustration we gave up for the day, but later that night I decided that doing it manually was the wrong way to handle it. As such I wrote some simple java code to do it for me. The code basically did exactly what I was doing above. Try the first 10 digits, if not a prime, drop the first digit and pick up another in the list, recording anything that was a 10 digit prime. I did this up to 720 decimal places, wanting to be sure to get it! I ended up with something like:

1. 7427466391

2. 7413596629

3. 6059563073

4. 3490763233 etc.

And bingo! Number 2 was my answer. I was so chuffed I thought nothing more of it and went to the next class feeling very pleased with myself. Turns out I had forgotten about the {}.com part. Another guy in the class pointed this out, so naturally we pointed our web browser at 7413596629.com and it took us to a website containing just:

f(1)= 7182818284

f(2)= 8182845904

f(3)= 8747135266

f(4)= 7427466391

f(5)= __________

Hours later we discovered that

7+1+8+2+8+1+8+2+8+4 = 49

8+1+8+2+8+4+5+9+0+4 = 49

8+7+4+7+1+3+5+2+6+6 = 49

7+4+2+7+4+6+6+3+9+1 = 49

Also we discovered that the 10 digit numbers above all were 10 digit numbers from the decimal of *e* . So clearly we just needed to find the 5th 10 digit number in *e* decimal places that added to 49. Code time again! The code was much the same, but this time it dropped the first number and picked up another if it wasn’t 49, not if it wasn’t a prime. This time my list was:

1. 7182818284

2. 8182845904

3. 8747135266

4. 7427466391

5. 5966290435

6. 2952605956

7. 0753907774

8. 0777449920 etc.

So 5966290435 was our answer!

We emailed this to the professor who simply replied “Very good”.

The next day he explained to us this was a recruiting tool used by Google. Turns out they put a billboard up in silicon valley trying to attract some young math’s brains to the company(see below). He modified it slightly so that we could use it, and successfully united the small class by having us attempt this challenge without his help.

Google is now renowned for their Google Labs Aptitude Test, which really puts potential employees through their paces, both mathematically and creatively, but I believe it all stemmed from this billboard.

Anyway here are some more brainteasers to keep you busy:

1. Five men dig five holes five feet deep in five hours. How long will it take one man to dig a half a hole?

2. Imagine an analog clock set to 12 o’clock. Note that the hour and minute hands overlap. How many times each day do both the hour and minute hands overlap? How would you determine the exact times of the day that this occurs?

3. There are 4 women who want to cross a bridge. They all begin on the same side. You have 17 minutes to get all of them across to the other side.

It is night. There is one flashlight. A maximum of two people can cross at one time. Any party who crosses, either 1 or 2 people, must have the flashlight with them. The flashlight must be walked back and forth, it cannot be thrown, etc. Each woman walks at a different speed. A pair must walk together at the rate of the slower woman’s pace.

Woman 1: 1 minute to cross

Woman 2: 2 minutes to cross

Woman 3: 5 minutes to cross

Woman 4: 10 minutes to cross

For example, if Woman 1 and Woman 4 walk across first, 10 minutes have elapsed when they get to the other side of the bridge. If Woman 4 then returns with the flashlight, a total of 20 minutes have passed and you have failed the mission.

What is the order required to get all women across in 17 minutes?

4. You’ve got someone working for you for seven days and a gold bar to pay them. The gold bar is segmented into seven connected pieces. You must give them a piece of gold at the end of every day.

If you are only allowed to make two breaks in the gold bar, how do you pay your worker?