Author |
Topic: Ladder and Crate (Part 2) (Read 1228 times) |
|
ThudnBlunder
wu::riddles Moderator
Uberpuzzler
The dewdrop slides into the shining Sea
Gender: 
Posts: 4489
|
 |
Ladder and Crate (Part 2)
« on: Jul 18th, 2008, 5:27pm » |
 Quote  Modify
|
A headmaster composed the following problem for his group of four high-flying maths students:
A long fireman's ladder of length d" rests squarely, and at an angle of less than 45o, against a vertical wall with its foot on the horizontal ground, a distance b" from the wall. A cubical crate fits flush into the angle of the wall and the ground, and just touches the ladder. What is the length of the side of the crate?
After some thought he then replaced the constants b and d with integers, such that the answer would also be an integer. However, to each student he gave a different integer value (< 1800) for d, the length of the ladder. The answers submitted by the four students were all the same. And they were all correct! So what was the side length of the crate?
|
|
 IP Logged |
THE MEEK SHALL INHERIT THE EARTH.....................................................................er, if that's all right with the rest of you.
|
|
|
Eigenray
wu::riddles Moderator
Uberpuzzler
Gender:
Posts: 1948
|
|
Re: Ladder and Crate (Part 2)
« Reply #1 on: Jul 18th, 2008, 6:27pm » |
Quote Modify
|
High-flying, eh? 
But wouldn't d  2008 be more appropriate?
|
| « Last Edit: Jul 18th, 2008, 6:31pm by Eigenray » |
IP Logged |
|
|
|
ThudnBlunder
wu::riddles Moderator
Uberpuzzler
The dewdrop slides into the shining Sea
Gender:
Posts: 4489
|
|
Re: Ladder and Crate (Part 2)
« Reply #2 on: Jul 18th, 2008, 7:17pm » |
Quote Modify
|
on Jul 18th, 2008, 6:27pm, Eigenray wrote:High-flying, eh?
|
|
Yeah, it's a Brit primary school.  But I am prepared to concede that you know the meaning of that word better than I do.
I can't' see why 2008 would be more appropriate.
Unless one is a numerologist (perish the thought), 1800 rules out the next possible solution just as well as 2008.
Or are you just letting me know that you have already solved it?  Perhaps I should have put it in Easy. 
|
| « Last Edit: Jul 18th, 2008, 7:40pm by ThudnBlunder » |
IP Logged |
THE MEEK SHALL INHERIT THE EARTH.....................................................................er, if that's all right with the rest of you.
|
|
|
Eigenray
wu::riddles Moderator
Uberpuzzler
Gender:
Posts: 1948
|
|
Re: Ladder and Crate (Part 2)
« Reply #3 on: Jul 18th, 2008, 8:49pm » |
Quote Modify
|
on Jul 18th, 2008, 7:17pm, ThudanBlunder wrote:| Yeah, it's a Brit primary school. But I am prepared to concede that you know the meaning of that word better than I do. |
|
I thought it was a reference to the answer. 
Quote:I can't' see why 2008 would be more appropriate.
Unless one is a numerologist (perish the thought), 1800 rules out the next possible solution just as well as 2008. |
|
It is the largest bound for which there is a unique solution, not to mention the current year.
Quote:| Or are you just letting me know that you have already solved it? |
|
What, me solve a math problem? You must be thinking of someone else.
Quote:| Perhaps I should have put it in Easy. |
|
Depends, is it meant to be done by hand?
|
| « Last Edit: Jul 18th, 2008, 8:49pm by Eigenray » |
IP Logged |
|
|
|
ThudnBlunder
wu::riddles Moderator
Uberpuzzler
The dewdrop slides into the shining Sea
Gender:
Posts: 4489
|
|
Re: Ladder and Crate (Part 2)
« Reply #4 on: Jul 18th, 2008, 9:24pm » |
Quote Modify
|
on Jul 18th, 2008, 8:49pm, Eigenray wrote:
It is the largest bound for which there is a unique solution, not to mention the current year. |
|
True; but I thought a 150' ladder was already pushing it.
Quote:| What, me solve a math problem? You must be thinking of someone else. |
|
Someone else? Well, I reckon anyone whose surname begins with a trigonometric function must be pretty good at maths.
Quote:| Depends, is it meant to be done by hand? |
|
Even if not, one still needs to derive the formulae.
|
| « Last Edit: Jul 23rd, 2008, 6:28pm by ThudnBlunder » |
IP Logged |
THE MEEK SHALL INHERIT THE EARTH.....................................................................er, if that's all right with the rest of you.
|
|
|
Eigenray
wu::riddles Moderator
Uberpuzzler
Gender:
Posts: 1948
|
|
Re: Ladder and Crate (Part 2)
« Reply #5 on: Jul 19th, 2008, 12:52pm » |
Quote Modify
|
Say the height of the ladder is a, so a2+b2=d2. Then
| hidden: | c = ab/(a+b), and a = bc/(b-c) is rational.
Since b,c are integers, a is an integer. So
a = k(m2-n2)
b = k(2mn)
d = k(m2+n2),
where n<m are relatively prime of opposite parity, and
c = ab/(a+b) = 2kmn(m2-n2)/[m2+2mn-n2]
is an integer.
But m2+2mn-n2 is relatively prime to 2mn(m2-n2), so
k = u(m2+2mn-n2)
for some integer u. Now
d = u(m2+2mn-n2)(m2+n2) < 1800.
Since m2+2mn-n2 = 2m2-(m-n)2 >= m2+2m-1,
m4-1 < (m2+2m-1)(m2+1) < 1800,
so m <= 6. So there are not very many possibilities.
For each pair (m,n), c = 2mn(m+n)(m-n)*u, where
u < 1800/[(m2+2mn-n2)(m2+n2)].
Here are the possibilities for (m,n), and c:
(2,1): c=12*{1,...,51}
(3,2): c=60*{1,...,8}
(4,1): c=120*{1,2,3,4}
(4,3): c=168*{1,2}
(5,2): c=420
(6,1): c=420
If c appears 4 times, it must be on one of the last 3 lists.
168 and 336 are only on 2 of the lists, so c=420. |
|
| « Last Edit: Jul 19th, 2008, 12:56pm by Eigenray » |
IP Logged |
|
|
|
ThudnBlunder
wu::riddles Moderator
Uberpuzzler
The dewdrop slides into the shining Sea
Gender:
Posts: 4489
|
|
Re: Ladder and Crate (Part 2)
« Reply #6 on: Jul 23rd, 2008, 6:50pm » |
Quote Modify
|
To sum up,
a = Max or Min {n(p2 - q2)(p2 - q2 + 2pq), 2npq(p2 - q2 + 2pq)}
b = Max or Min {n(p2 - q2)(p2 - q2 + 2pq), 2npq(p2 - q2 + 2pq)}
c = 2npq((p2 - q2)
d = n(p2 + q2)(p2 - q2 + 2pq)
for integers n,p,q
|
| « Last Edit: Jul 24th, 2008, 4:34pm by ThudnBlunder » |
IP Logged |
THE MEEK SHALL INHERIT THE EARTH.....................................................................er, if that's all right with the rest of you.
|
|
|
|
RIDDLES SITE
WRITE MATH!
Home 
Help 
Search 
Members 
Login 
Register
Reply
Notify of replies
Send Topic
Print