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NeoPets Riddles (Lenny Conundrums) and Answers Here

 
 
escabatum
 
  1  
Reply Thu 18 May, 2006 12:14 am
how I solved it
I solved it this way:
since if the farmer with 14 gives three each to farmer with 9 and farmer with 13, the equation would be:
14a - 6 = 13b +3 = 9c + 3. So since 13b + 3 = 9c +3, if you subtract 3 from each side you get 13b = 9c, so you just figure out what value would make those equal, and the easiest is to blug in the other - so 13(9) = 9(13) = 120. Then you check it with all the other answers by setting 120 equal to all the other equations. I got 1800 too. Hope I got in the first 250!
0 Replies
 
soviet1918
 
  1  
Reply Thu 18 May, 2006 01:18 pm
Laughing Well i got 1800 BUT at school i failed maths soooo Crying or Very sad
0 Replies
 
scenceable
 
  1  
Reply Thu 18 May, 2006 04:25 pm
hey I got the answer last time right but...I didn't get anything. was I too slow? they say all correct answers get something. hmmmm
0 Replies
 
stapel
 
  1  
Reply Thu 18 May, 2006 09:10 pm
Round 167
The 'Lenny' authors, in Round 167, wrote:
Fifteen different farmers each had their own apple orchards. The first farmer had only one apple tree in his orchard, the second farmer had two trees in his orchard, and so on up to fifteen trees for the fifteenth farmer.

At harvest time, each farmer noticed something very peculiar: each tree in the same orchard produced the exact same number of apples. (Note that the number of apples per tree varied from orchard to orchard) And when all fifteen farmers got together to talk about it, they all realized that, if the farmer with eleven trees had given one apple to the farmer with seven trees, and the farmer with fourteen trees had given three apples each to the farmer with nine trees and the farmer with thirteen trees, they'd all end up with the exact same number of apples!

How many apples did all the farmers produce, in total?

Define a, b, c, d, and e as follows:

. . .a: number of apples per tree in 7th orchard

. . .b: number of apples per tree in 9th orchard

. . .c: number of apples per tree in 11th orchard

. . .d: number of apples per tree in 13th orchard

. . .e: number of apples per tree in 14th orchard

Then we have:

. . .11c - 1 = 7a + 1

. . .14e - 6 = 9b + 3

. . .14e - 6 = 13d + 3

. . .. . .7a - 11c = -2

. . .9b - 14e = -9

. . .13d - 14e = -9

. . .11c - 14e = -5

Adding the first and fourth equations, we get the following reduced system:

. . .7a - 14e = -7

. . .9b - 14e = -9

. . .13d - 14e = -9

Note that the last two equations require that 9b = 13d.

Subtracting the third equation from the second and first equations, we further reduce the system and get:

. . .7a - 13d = 2

. . .9b - 13d = 0

Solving the above (under-determined) system, we get:

. . .a = (13/7)d + 2/7

. . .b = (13/9)d

For this to be a sensible system, we must have the right-hand sides come out to whole numbers. This confirms that d is a multiple of 9 and that b is a multiple of 13. Start testing numbers with the first multiple of 9:

. . .. . .d = 9: (13/9)d = 13 = b

Also, if b = 13, then there are 9(13) + 3 = 117 + 3 = 120 apples in each orchard (after the farmers make their adjustments). So try "d = 9" as a solution for the entire system. If d = 9, then:

. . .a = (13/7)(9) + 2/7 = 117/7 + 2/7 = 119/7 = 17

. . .b = 13

Since 7a - 11c = -2, then 11c = 7a + 2, and:

. . .c = (7a + 2)/11 = (119 + 2)/11 = 121/11 = 11

. . .d = 9 (by assumption)

Since 11c - 14e = -5, then 14e = 11c + 5, and:

. . .e = (11c + 5)/14 = (121 + 5)/14 = 126/14 = 9

Therefore, we have:

. . .farmer 7 has 7(17) + 1 = 119 + 1 = 120 apples

. . .farmer 9 has 9(13) + 3 = 117 + 3 = 120 apples

. . .farmer 11 has 11(11) - 1 = 121 - 1 = 120 apples

. . .farmer 13 has 13(9) + 3 = 117 + 3 = 120 apples

. . .farmer 14 has 14(9) - 6 = 126 - 6 = 120 apples

So the solution checks.

However, this is, as was noted earlier, an underdetermined system. The generalized solution is of the form:

. . .(a, b, c, d, e) = (2e - 1, (14/9)e - 1, (14/11)e - 5/11, (14/13)e - 9/13, e)

That is, any value of e which returns whole-number values for a, b, c, and d, and for which 14e - 6 is a multiple of 120, will work. For instance:

. . .e = 25749
. . .a = 51497
. . .b = 40053
. . .c = 32771
. . .d = 27729

This yields a per-orchard value of 360480 = 3004(120), and a total value of 5407200.

It is reasonable to assume, however, that the smaller total value of 1800 is the expected answer.

Eliz.
0 Replies
 
escabatum
 
  1  
Reply Thu 18 May, 2006 09:10 pm
no prize
Did you check your neomail, it would have said you got about 200 or 300 np, thats how much I got for the Geraptiku one even though I wasn't one of the first 250. No plushie or trophie for me... =(
0 Replies
 
soviet1918
 
  1  
Reply Fri 19 May, 2006 01:52 pm
3 or 4 days
yeah to get an answer from the TNT can take 3 or 4 days..
0 Replies
 
quietStorm7
 
  1  
Reply Fri 19 May, 2006 02:26 pm
yea i know it would
0 Replies
 
scenceable
 
  1  
Reply Fri 19 May, 2006 07:20 pm
wow I'm smart. it doesn't notify you (like, give you that envelope thing in the top right corner) for the neomail, but its there.

Very Happy
0 Replies
 
Fruitbat
 
  1  
Reply Mon 22 May, 2006 06:20 pm
I think stapel has hit the nail on the head.
There are infinitely many solutions. I won't bore you with the full calculation (it has been suggested above). The gist of it is - if you assume nothing other than what you have been told, then you deduce that each farmer has 120y apples for some y. A little more deduction (considering what must be divisible by what) gives (y-1) =3003z for some z. The answer we seem to be agreed on (1800) corresponds to z=0, y=1. However, unless I've made a monumental boob in my calculation, all positve integer values of z will work. Now, you could argue about what is a sensible number of apples in a tree, but that's not really relevant in a mathematical puzzle. I think they've boobed this week.
Anyone see a flaw in my logic?
0 Replies
 
soviet1918
 
  1  
Reply Tue 23 May, 2006 02:08 pm
Agree
The answer must be 1800 because thats the answer most people have come up with,,we cant all be wrong.... Laughing can we?
And ive still not herd fron TNT if ive won or not,or do they contact you on the thursday some time when it closes? my answer was posted about an hour after the new Conundrum came out..
0 Replies
 
stapel
 
  1  
Reply Tue 23 May, 2006 04:21 pm
soviet1918 wrote:
The answer must be 1800 because thats the answer most people have come up with

Too bad the correct answer isn't determined by popularity.

As previously mentioned, "1800" is likely the expected answer but, as proven mathematically, it is not by any means the only answer. Since the "right" answer is the expected one, probably "1800" is "right", but it is not "the" answer, as far as the math is concerned.

Eliz.
0 Replies
 
purplemonkey
 
  1  
Reply Tue 23 May, 2006 09:14 pm
Re: Agree
soviet1918 wrote:
And ive still not herd fron TNT if ive won or not,or do they contact you on the thursday some time when it closes?


yea they send out the neomails on wednessday when the news gets posted to the winners.
0 Replies
 
markr
 
  1  
Reply Wed 24 May, 2006 09:40 am
Fruitbat wrote:
Now, you could argue about what is a sensible number of apples in a tree, but that's not really relevant in a mathematical puzzle.


It most certainly is relevant (it's not purely a math problem - apple trees don't exist in mathematics), and the next smallest solution yields an unrealistic number.
0 Replies
 
melodyrose27
 
  1  
Reply Wed 24 May, 2006 11:14 am
Congratulations! You have guessed correctly in the Lenny Conundrum game (round 167). You have won 386 NP!

Yours Sincerely,
The Neopets Team!
0 Replies
 
Fruitbat
 
  1  
Reply Wed 24 May, 2006 11:55 am
markr wrote:
Fruitbat wrote:
Now, you could argue about what is a sensible number of apples in a tree, but that's not really relevant in a mathematical puzzle.


It most certainly is relevant (it's not purely a math problem - apple trees don't exist in mathematics), and the next smallest solution yields an unrealistic number.


Apple trees can exist in mathematics. You just need to devise an axiomatic definition of them. A "tree" is a container for "apples". If you start bringing in "real world" rules like "realistic numbers" of apples, then where do you stop? Do we actually know the maximum number of apples that a tree can produce? How likely is it that an orchard of 15 trees would produce the same number of apples per tree?

In this context (mathematical puzzle), these are not real apple trees. They are abstract constructs and as such can contain any number of "apples" (again abstract) unless defined to have an upper bound.
0 Replies
 
perilette
 
  1  
Reply Wed 24 May, 2006 09:03 pm
New Lenny Puzzle
What is the next number in this sequence?

22272
24742
25862766
2674


I've always hated numbers.
0 Replies
 
baconNtofu
 
  1  
Reply Wed 24 May, 2006 09:22 pm
22272
24742
25862766
2674

the answer is 7150276 Smile
0 Replies
 
lem
 
  1  
Reply Wed 24 May, 2006 09:24 pm
baconNtofu wrote:
22272
24742
25862766
2674

the answer is 7150276 Smile


How did you come up wit it?
0 Replies
 
turkishharem
 
  1  
Reply Wed 24 May, 2006 09:24 pm
How did you get that?
0 Replies
 
baconNtofu
 
  1  
Reply Wed 24 May, 2006 09:26 pm
hmmmm i was about to explain my answer, but i think it may be wrong :\ i need to go find me a calculator.
0 Replies
 
 

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