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ThatDonGuy
ThatDonGuy
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Gialmere
May 16th, 2022 at 7:36:20 AM permalink

Assuming I am reading this right, the number of bridges does not matter.
If you start with 2 cakes, each troll will take 1 cake, then give it back.

The answer is 2 cakes



The train moves its length - 30 feet in the time it takes Al to walk 30 feet
The train moves its length + 45 feet in the time it takes Ben to walk 45 feet
The train moves 75 feet in the time it takes either to walk 15 feet, so it moves 150 feet in the time it takes Al to walk 30 feet
150 feet = its length - 30 feet, so the train's length is 180 feet

charliepatrick
charliepatrick
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Gialmere
May 16th, 2022 at 7:55:11 AM permalink
Quote: Gialmere

Here's a few easy Monday puzzles...



You are on your way to visit your Grandma, who lives at the end of the valley. It's her birthday, and you want to give her the cakes you've made.

Between your house and her house, you have to cross 7 bridges, and as it goes in the land of make believe, there is a troll under every bridge! Each troll, quite rightly, insists that you pay a troll toll. Before you can cross their bridge, you have to give them half of the cakes you are carrying, but as they are kind trolls, they each give you back a single cake.

How many cakes do you have to leave home with to make sure that you arrive at Grandma's with exactly 2 cakes?
link to original post

What a great puzzle...
After the last bridge you have to have two cakes, The troll gives you one, so after the troll had taken their toll you needed one cake. Hence prior to the bridge you had two cakes.

So at each bridge you arrive with two cakes, pay one cake in toll, and receive one back. Thus you can start with two cakes!

No need to work through seven iterations!
charliepatrick
charliepatrick
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Gialmere
May 16th, 2022 at 8:05:01 AM permalink
Let's assume the walkers travel at 15 ft/sec and the train is travelling right to left. After 2 seconds the back of the train is 30 feet from the starting point (on the right) as that is how far Al has walked, call this "X". In the next second Ben manages to complete his 45ft walk so is 45 feet to the left of the starting point, call this "Y"; at this time the back of the train passes him. So adding 45ft and 30ft gets the distance (YX) and is how far the back of the train travelled in the third second. Thus the train travels 75ft/sec.

We also know in the first two seconds the back of the train reached 30ft from the starting position, hence it was 30+2*75 foot long = 180ft.
Dieter
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Dieter
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May 16th, 2022 at 8:29:58 AM permalink
Only 2 cars long?


Boss told me that as a security guard, it's my job to watch the office. I'm up to season 4.
May the cards fall in your favor.
ThatDonGuy
ThatDonGuy
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May 16th, 2022 at 9:07:39 AM permalink
Quote: Dieter

Boss told me that as a security guard, it's my job to watch the office. I'm up to season 4.
link to original post


Don't be ridiculous - The Office only had 2 seasons (and a Christmas special).
Oh, you're thinking about that American nonsense, aren't you?
Wizard
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Wizard
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Gialmere
May 16th, 2022 at 10:54:09 AM permalink
RE: Puzzle #1

I came up with the same answer everyone else did.

May I suggest an alternate version where grandma gets 3 cakes?

Is there a way to express the answer as a function of c cakes and t trolls?
It's not whether you win or lose; it's whether or not you had a good bet.
ThatDonGuy
ThatDonGuy
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May 16th, 2022 at 11:26:19 AM permalink
Quote: Wizard

RE: Puzzle #1

I came up with the same answer everyone else did.

May I suggest an alternate version where grandma gets 3 cakes?

Is there a way to express the answer as a function of c cakes and t trolls?
link to original post



Assuming C is the number of cakes you have at the end, you start with 2^T C - 2^(T + 1) + 2 cakes

If you work backwards, and assume you have 2^T C - 2^(T + 1) + 2 cakes with T bridges remaining, then, at the previous bridge, you had 2^T C - 2^(T + 1) + 1 before you got one back, which meant you had twice as many, or 2^(T + 1) C - 2^((T + 1) + 1) + 2, when you reached that bridge.

Gialmere
Gialmere
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May 16th, 2022 at 5:29:53 PM permalink
Quote: ThatDonGuy


Assuming I am reading this right, the number of bridges does not matter.
If you start with 2 cakes, each troll will take 1 cake, then give it back.

The answer is 2 cakes



The train moves its length - 30 feet in the time it takes Al to walk 30 feet
The train moves its length + 45 feet in the time it takes Ben to walk 45 feet
The train moves 75 feet in the time it takes either to walk 15 feet, so it moves 150 feet in the time it takes Al to walk 30 feet
150 feet = its length - 30 feet, so the train's length is 180 feet


Quote: charliepatrick

After the last bridge you have to have two cakes, The troll gives you one, so after the troll had taken their toll you needed one cake. Hence prior to the bridge you had two cakes.

So at each bridge you arrive with two cakes, pay one cake in toll, and receive one back. Thus you can start with two cakes!

No need to work through seven iterations!

Let's assume the walkers travel at 15 ft/sec and the train is travelling right to left. After 2 seconds the back of the train is 30 feet from the starting point (on the right) as that is how far Al has walked, call this "X". In the next second Ben manages to complete his 45ft walk so is 45 feet to the left of the starting point, call this "Y"; at this time the back of the train passes him. So adding 45ft and 30ft gets the distance (YX) and is how far the back of the train travelled in the third second. Thus the train travels 75ft/sec.

We also know in the first two seconds the back of the train reached 30ft from the starting position, hence it was 30+2*75 foot long = 180ft.


Quote: Wizard

RE: Puzzle #1

I came up with the same answer everyone else did.


Correct!!

And so children, the moral of the story is...

If you don't know math, you'll starve to death.

Yeah, it is a short train. (For the record, a boxcar is just over 55' in length). Perhaps it's a Metrolink, or maybe a kiddie ride at an amusement park.

------------------------------------------------------

They get a better signal.

Will they expect me to know the job, or will they train me?
Have you tried 22 tonight? I said 22.
Gialmere
Gialmere
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May 17th, 2022 at 7:34:31 AM permalink
A puzzle inspired by Catriona Shearer...



The picture above shows two semicircles. The lighter region (inside the larger semicircle but outside the smaller one) has an area of 7.

What’s the area of the darker region?
Have you tried 22 tonight? I said 22.
charliepatrick
charliepatrick
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Gialmere
May 17th, 2022 at 7:54:29 AM permalink
Quote: Gialmere

...What’s the area of the darker region?
link to original post

It's also 7.

The logic involves creating right angled triangles (one for each semi circle) that can be formed used the diameter as the hypotenuse. Consider the large one first, its hypotenuse is 2r. Now chop the triangle in half (so two of the sides of it are radii) - clearly its area is half the original one. Then rotate it by 45 degrees. This will now line up so its hypotenuse matches the diameter of the smaller semi-circle.

Thus by similar triangles within similar semi circles, the area of the smaller semi-circle is half the larger one.

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