Brain Teasers and Maths Puzzles

[eMTe]

Yeah, sexism is gay.

[Tormuse]

Also, the parents left their kids alone three times in that solution. How irresponsible!

I can't think of any puzzles right now, so if anyone else wants to volunteer one, go ahead.

[eMTe]

Once upon a time, the Union College administration was taken over by a rogue band of number theorists intent on developing a new system of student ID numbers. The number theorists wanted the new ID numbers to have ten digits in which each of the numerals from 0 to 9 appeared exactly once. They also wanted each ID number to be divisible by each of the digits (except 0!).

Questions

1. What would be the smallest possible new ID number?

2. What would be the largest possible new ID number?

[Tormuse]

So, someone in RL told me another lateral thinking puzzle and I thought, "I have to post this on CWF!" and I brought up this thread and said, "Oh yeah, the number puzzle... I guess I'd better take a stab at this one first..."

So, any number that has digits that add up to a multiple of 3 is divisible by 3 and any number with digits adding up to a multiple of 9 is divisible by 9. Luckily, the digits from 1 to 9 add up to 45, which is divisible by 3 and 9, so those digits are taken care of.

Any number divisible by 5 ends in either a 5 or a 0. Since we want our result to be even, (to be divisible by 2) all our IDs are going to end in 0. This also takes care of 6, because 2 and 3 are the factors of 6 and if a number is divisible by those two numbers, then it's divisible by 6.

I've also heard the rule that if the last two digits are divisible by 4, the whole number is divisible by 4 and if the last three digits are divisible by 8, the whole number is divisible by 8. This means that the second-last digit must be even. If the third-last digit is odd, the second-last is 2 or 6, and if the third-last is even, the second-last is 4 or 8.

It goes without saying that all possible IDs are divisible by 1 and you don't even want to think about dividing by zero!



So, that's all digits taken care of except for 7... and now, I've run out of tricks, so the only thing I can think to do is to keep adding 2520 (5*7*8*9 to get the smallest possible number that's divisible by all the digits) over and over again until I get a number that has all the digits in it... (In retrospect, this makes all the calculations I did above completely moot, but let's not think about that)

And now, I've run out of time and have to go to work shortly, so I'll have to finish this later...

[Tormuse]

Okay, continuing from above... Again, I couldn't figure out any trick for determining if a number is divisible by 7, so I had to resort to a bit of brute force to figure out the answer:

I used Microsoft calculator to help me and started with the number 9876543210. Then, I repeatedly subtracted 90 from it until I had a number divisible by 2520. After that, I just kept subtracting 2520 over and over again (That is I subtracted 2520 once and then pushed Enter over and over) until I arrived at a number that contained all the digits. The first one I arrived at was 9876351240, so that's the largest number.

To find the smallest number, I did the same thing in reverse: I started with 1234567890, added 90 until it was divisible by 2520 and then added 2520 until it contained all the digits. The first one I arrived at was 1234759680, so that's the smallest number.

TL;DR:

1) Smallest number: 1234759680

2) Largest number: 9876351240

Please tell me that's the right answer, eMTe.

[Zyx]

I used Microsoft calculator to help me and started with the number 9876543210.

You did not have a copy of Excel anywhere nearby? You'll probably start to cry when I tell you about its functions GCD and LCM.

[Tormuse]

Darn, I didn't even think of Excel! Oh well, live and learn.

[Drasir-Vel]

There are two triangles.

Triangle one:
Side A: 5cm
Side B: 5cm
Side C: 6cm

Triangle two:
Side A: 5cm
Side B: 5cm
Side C: 8cm

Which of these triangles has the greater area?

[Tormuse]

Oh, are we going to the next puzzle already? I thought we were waiting for eMTe to confirm that I got it right! Well, anyway, here goes...

I vaguely remember my high school math class including some way of calculating the area of triangles, and I think it might have involved sine, cosine, and tangent calculations or something, but it's all a blur, so I'll see if I can figure something else out...

Both triangles are isoceles, (2 equal sides) which means that they are symmetrical, so if I draw a line down the middle, I will get two equally sized right-angled triangles like so:

Code: Select all

    /|\
   / | \
  /  |  \
 /___|___\

(Kinda reminds me of the Atari logo)

Now, for right-angled triangles, I can apply the Pythagorean theorem which I *do* remember from math class!

Triangle one gets divided into two triangles with a hypotenuse of 5 and other sides 3 and unknown.

Unknown side = root of (5 squared - 3 squared) = root of (25 - 9) = root of 16 = 4

Now, if we imagine one of those two right-angled triangles flipped and moved on to the other side, like so...

Code: Select all

_____
|   /
|  /|
| / |
|/__|

(excuse my terrible ascii art) ...they form a rectangle with sides 3 and 4 with an area of 12.

Do the same for triangle two and you get two right-angled triangles with hypotenuse 5 and other sides 4 and unknown.

Unknown side = root of (5 squared - 4 squared) = root of (25 - 16) = root of 9 = 3

Doing the same trick with flipping one triangle and moving it to the other side gets us the same sized rectangle with sides 3 and 4 with an area of 12.

TL;DR: Triangle one and Triangle two have the same area!

[Drasir-Vel]

That's right. You'd imagine the one with the higher number would be bigger, but that's not how it works, because then at least one other side would have to be bigger too.
Sorry for jumping to the next puzzle before eMTe's was finished.

[Tormuse]

That's okay. Here's mine:

Julie has just been murdered... She's lying in bed in her room. There is a puddle of water on the floor. The murder weapon is a pair of scissors, but there isn't a single cut or scratch on her.

How did she die?

[Zyx]

How did she die?

Alone. Forgotten. It was a tragic death, really. The cops ruled it out as a suicide. But I could not believe that, it wasn't at all what Julie would have done. There was no reason for her suicide. Everything was going up in her life, a new job and a new boyfriend, both that she loved dearly. They never got to go on their dream trip to Cambodia to see Angkor Wat. It is the largest Hindu temple complex in the world. The temple was built by King Suryavarman II in the early 12th century in Yasodharapura (Khmer: យសោធរបុរៈ, present-day Angkor), the capital of the Khmer Empire, as his state temple and eventual mausoleum. Breaking from the Shaivism tradition of previous kings, Angkor Wat was instead dedicated to Vishnu. As the best-preserved temple at the site, it is the only one to have remained a significant religious centre since its foundation – first Hindu, dedicated to the god Vishnu, then Buddhist. The temple is at the top of the high classical style of Khmer architecture. It has become a symbol of Cambodia, appearing on its national flag, and it is the country's prime attraction for visitors.

Angkor Wat combines two basic plans of Khmer temple architecture: the temple mountain and the later galleried temple, based on early Dravidian Architecture, with key features such as the Jagati. It is designed to represent Mount Meru, home of the devas in Hindu mythology: within a moat and an outer wall 3.6 kilometres (2.2 mi) long are three rectangular galleries, each raised above the next. At the centre of the temple stands a quincunx of towers. Unlike most Angkorian temples, Angkor Wat is oriented to the west; scholars are divided as to the significance of this. The temple is admired for the grandeur and harmony of the architecture, its extensive bas-reliefs, and for the numerous devatas adorning its walls.

Nah, the scissors were used to cut a rope holding a massive block of ice that crushed her.

[Drasir-Vel]

I could imagine the scissors cut a tube which supplied water to her, which she for some reason needed. Maybe because she was in a coma, and couldn't drink herself.

But why would she be at home in a state like that. Most likely if she was like that, you would want to keep her at the hospital. Maybe she actually was at the hospital. "Her room" was actually her room in the hospital.

But that doesn't explain why nobody had noticed it before she died. The nurses would be checking up on her.

[Tormuse]

How did she die?

Alone. Forgotten. It was a tragic death, really. The cops ruled it out as a suicide. But I could not believe that, it wasn't at all what Julie would have done. There was no reason for her suicide. Everything was going up in her life, a new job and a new boyfriend, both that she loved dearly. They never got to go on their dream trip to Cambodia to see Angkor Wat. It is the largest Hindu temple complex in the world. The temple was built by King Suryavarman II in the early 12th century in Yasodharapura (Khmer: យសោធរបុរៈ, present-day Angkor), the capital of the Khmer Empire, as his state temple and eventual mausoleum. Breaking from the Shaivism tradition of previous kings, Angkor Wat was instead dedicated to Vishnu. As the best-preserved temple at the site, it is the only one to have remained a significant religious centre since its foundation – first Hindu, dedicated to the god Vishnu, then Buddhist. The temple is at the top of the high classical style of Khmer architecture. It has become a symbol of Cambodia, appearing on its national flag, and it is the country's prime attraction for visitors.

Angkor Wat combines two basic plans of Khmer temple architecture: the temple mountain and the later galleried temple, based on early Dravidian Architecture, with key features such as the Jagati. It is designed to represent Mount Meru, home of the devas in Hindu mythology: within a moat and an outer wall 3.6 kilometres (2.2 mi) long are three rectangular galleries, each raised above the next. At the centre of the temple stands a quincunx of towers. Unlike most Angkorian temples, Angkor Wat is oriented to the west; scholars are divided as to the significance of this. The temple is admired for the grandeur and harmony of the architecture, its extensive bas-reliefs, and for the numerous devatas adorning its walls.

No, she was planning to go to England to see Stonehenge, an ancient circle of stones that archaeologists believe was built anywhere between 3000 and 2000 BC and is now a world heritage site. (Just kidding)

Nah, the scissors were used to cut a rope holding a massive block of ice that crushed her.

No. Good guess, though.

I could imagine the scissors cut a tube which supplied water to her, which she for some reason needed. Maybe because she was in a coma, and couldn't drink herself.

I like this guess even better. It's still not right, though.

But why would she be at home in a state like that. Most likely if she was like that, you would want to keep her at the hospital. Maybe she actually was at the hospital. "Her room" was actually her room in the hospital.

But that doesn't explain why nobody had noticed it before she died. The nurses would be checking up on her.

Well, lateral thinking puzzles don't always make complete logical sense. I think it's fair to say that she isn't at a hospital, though. She is in her bedroom at home.

[Zyx]

No, she was planning to go to England to see Stonehenge, an ancient circle of stones that archaeologists believe was built anywhere between 3000 and 2000 BC and is now a world heritage site. (Just kidding)

Ah, I ruled out Stonehenge because I thought the puddle of water was from a recent trip to England (where, as everyone knows, it always rains).

[eMTe]

Please tell me that's the right answer, eMTe.

I tell you that's the right answer.

I think that important observation regarding the Julie puzzle is that she lies in bed while the puddle is on the floor.

[Tormuse]

Ah, I ruled out Stonehenge because I thought the puddle of water was from a recent trip to England (where, as everyone knows, it always rains).

Naw, she's actually quite fond of water; it just backfired on her this time.

I think that important observation regarding the Julie puzzle is that she lies in bed while the puddle is on the floor.

Actually, there's water all around and on the bed.

[eMTe]

My conclusion is that she was ill and received some kind of treatment at home, including being connected to the drip. Somebody cut the delivery cable and she died.

[Drasir-Vel]

Something that's somewhat related to a puzzle.

http://buttersafe.com/2007/11/15/the-brothers/

[Tormuse]

My conclusion is that she was ill and received some kind of treatment at home, including being connected to the drip. Somebody cut the delivery cable and she died.

No, she was in good health.