Lenny Conundrum #86

maruka · **Posts:** 567 **Joined:** Fri Jul 02, 2004 6:34 am

genie_mcbeal wrote:

I agree with yptee, the sinking is need not be taken into consideration. And I got a similar range of answer to him.

Hint:
If there are 4 rows, total number of marbles should be 1 + (1+2) + (1+2+3) + (1+2+3+4).

Objective is to find the number of rows, but u gotta use number of marbles to help u.

Eg, if there are only 10 kikos, there are 20 marbles, that means they can only stack up to 4 rows, meaning it would be 4 x 2 = 8 cm high

I tried to solve it by using a double summation at first, summation of the first n terms followed by the summation of that result from 1 to m. In other words, a summation (from 1 to m) of the summation of the first n terms. This answer must be less than or equal to the number of (kikos x 2). From there, solve the cubic equation and find the nearest integer m that satisfies the inequality.

Haha... Anyone even understand what I'm saying? Graduated with a degree in maths, but honest, u don't really need university knowledge of maths to do this. High School is enough.

On a last note, I tried using summation as mentioned earlier, but I couldn't solve the cubic equation because the number was too big. Neither did I use any maths algebraic software to help me with it. I simply took the advice of one of the hints earlier to help me out.

I got the overall summation as ½∑(n+n²), where n= numbers of layers. Hope this helps.... Ans is between 300 and 500.

weirdguy · **Posted:** Sat Oct 02, 2004 8:18 am

So I figure they use the "(out of 1,050,000)" number instead of the actual one because what if someone made another one? Wouldn't that completely throw everyone off?

eeevil · **Posts:** 48 **Joined:** Tue Aug 17, 2004 3:39 am

Quote:

I got the overall summation as ½∑(n+n²), where n= numbers of layers. Hope this helps.... Ans is between 300 and 500.

yeah, I got the same thing - then I wrote a small c-program to find out what number n actually is... I hope it's right. if my solution is right, it doesn't matter if you take the max number of kikos or the actual number of kikos; the max number of layers is the same.

*hopes her answer is right*

matchbow · **Posted:** Sat Oct 02, 2004 11:35 am

If there are marbles left over, should i just leave them out or add them to the problem somehow?

yptee · **Posted:** Sat Oct 02, 2004 11:52 am

weirdguy wrote:

So I figure they use the "(out of 1,050,000)" number instead of the actual one because what if someone made another one? Wouldn't that completely throw everyone off?

Actually it doesn't matter you use the actual or 1.05m, the answer is still the same.

In my calculation, I ignore any left over as it cannot be used to increase the height of the pyramid any further. I hope that helps.

artemisanoush · **Posted:** Sat Oct 02, 2004 12:25 pm

Finally!

The numbers 1, 3, 6, 10,....have the nth term given by n(n+1)/2

So the sum of n terms of the series is

n n
SUM[r(r+1)/2] = (1/2).SUM[r^2 + r]
r=1 r=1

= (1/2)[n(n+1)(2n+1)/6 + n(n+1)/2]

n(n+1)
= ------[2n+1 + 3]
12

= n(n+1)(2n+4)/12

= n(n+1)(n+2)/6

Check: If n=4 this gives 4 x 5 x 6/6 = 20 which is correct.

So we nust solve the equation

n(n+1)(n+2)/6 = 2,100,000

n(n+1)(n+2) = 12,600,000

The height of the pyramid is (Will be given to those who wish to be spoiled neomail artemisanoush)

genie_mcbeal · **Posts:** 6 **Joined:** Thu Jul 01, 2004 11:07 pm

The only problem is: this cubic equation is too difficult to solve without using computational means. So it will take either a maths software, a graphing software, write a program or use excel. In any case, I think puzzle is kinda difficult, it's purely mathematical and not many people can solve it unless they understand the concept of summation i think.

Well, but I like this kinda puzzles, at least there's logic and computable. Unlike the previous ones which did not have much mathematical logic involved, just a matter of pattern observation

Cranberry · **Posted:** Sat Oct 02, 2004 4:41 pm

You have to use every marble; the puzzle says so. "Every kiko in Neopia uses its marbles." And it asks "how many cm tall can the pyramid get?" (instead of "how many cms tall is the finished pyramid?"), which suggests it's not completed to a point, but still a work in progress when they run out of marbles. You just need to know how many cms they can manage with what they have.

stampsyne · **Posted:** Sun Oct 03, 2004 5:34 am

Brain.. h .. u .. r .. t .. s :roflol:

okamotosan18 · **Posted:** Mon Oct 04, 2004 5:52 am

Well, depending on if they wanted us to assume that all the Kikos were only the ones they created for us, then the number is 2cm less than the actual number of Kikos. Using the fact that they used all of their marbles, and assuming they tried to build the biggest pyramid they could, then the height was around 400cm, but unless you do the work for yourself, then that number is useless to you

artemisanoush · **Posted:** Wed Oct 06, 2004 11:48 pm

Congratulations! You have guessed correctly in the Lenny Conundrum game (Round 86). We have given you 1887 NP!

Woohoo! That's right baby.

gemma58 · **Posted:** Thu Oct 07, 2004 3:03 am

Congratulations! You have guessed correctly in the Lenny Conundrum game (Round 86). We have given you a Orange Chocolate Kiko, an Avatar, and 1887 NP! AND Third place Trophy at Lenny Conundrum!!

I answered 462 cm (231 layers x 2cm)

this site helped: http://milan.milanovic.org/math/Math.php?akcija=SviTetra

note: In this list the actual "layer" was 233 for the total number of kikos x 2 marbles
but as can be seen in this list the first two "layers" contain no total value so the "right" layer is 231.
Also it did not matter whether the kikos were calculated as 1050000 or 1050311 as both satisfy this answer.

Cranberry · **Posted:** Thu Oct 07, 2004 3:28 am

I won, too. I thought we'd get more than 1887 NP! I found this one much harder than the last one, but fewer people solved that one. Someone must have leaked this one's answer on the neoboards or something.

Also, looks like the kikos DID have to build the pyramid to a point and had marbles left over. I thought for sure I'd get it wrong and that 392 cm would be the answer; that's what it would have been if you had to use every marble (although the pyramid would not be complete; they'd get to 196 rows before running out of marbles).

gemma58 · **Posted:** Fri Oct 08, 2004 2:16 am

They completed 231 layers with 2081156 marbles and would not have sufficient marbles to complete the 232nd layer which would take 2108184 marbles (~ 8000 marbles short)

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Lenny Conundrum #86

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