33 Pearls Problem

Date: 11/14/96 at 20:19:53
From: James Fletcher
Subject: 33 Pearls Problem

Dear Dr. Math,

I'm having a hard time with this problem, harder than any word problem 
I've ever done before, and I think it is because I do not understand 
the concept behind it. The problem goes like this:

On a string of 33 pearls, the middle pearl is the largest and most  
expensive of all. Starting from one end, each pearl is worth $100 more  
than the one before, up to the middle. From the other end, each pearl 
is worth $150 more than the one before, up to the middle. The string 
of pearls is worth $65,000. What is the value of the middle pearl?

Thank you very much!

   James Fletcher

Date: 12/11/96 at 21:59:16
From: Doctor Daniel
Subject: Re: 33 Pearls Problem

Hi James,

You're right that this isn't an easy problem, so we'll take it slow.  
We'll start with some basics. For starters, there are exactly 16 
pearls to either side of the pearl in the center.  

We'll call the end where each pearl is worth $100 more than the 
previous one the left side, and the end where each pearl is worth $150 
more than the previous one the right side. (That's just so it's easier 
for me to write about them.) So I've cut the necklace into 3 parts: 
the left side, the middle pearl, and the right side. Each of the two 
sides has 16 pearls.  

Say that the left side has value L, the middle pearl has value M, and 
the right side has value R. Then we know that 

L + M + R = 65000.  

It'd be great if we could solve for L and for R in terms of M. Then 
we could solve for M, and we'd be done, right?

Well, what's L? We know that the pearl in the left side closest to 
the middle has value M-100. And the next one has value M-200, on to 
the last pearl which has value M-1600 (the easy way to convince 
yourself of this is to draw a picture of the whole necklace). So we 
have: 

L = (M-100) + (M-200) + (M-300) + ... + (M-1600).

I'm going to collect terms here to get the M's separate:

L = 16M - (100+200+300+...+1600)
L = 16M - 100(1+2+3+...+16)

Now, what's 1+2+...+16?  You could show, just by brute force, that 
this is 136. But that's unsatisfying, so let's take a quick detour.  

1+2+3+...+16 is the 16th triangle number. The general form for the nth 
triangle number is 1/2 * (n) * (n+1). Why?  Group the sum instead as

1+16  +  2+15  +  3+14  +  4+13  + . . . 

Each of these small blocks will have sum equal to 17 (In the general 
case, it will have sum n+1) and there will be exactly 8 of them (in 
the general case, n/2).  So the sum will be (n+1) * n/2, which is 
another way of stating that formula I just gave you.

So back to our word problem:

L = 16M - 100(1+2+...+16)
L = 16M - 100(17)(16/2)
L = 16M - 100*136
L = 16M - 13600.

Now let's look at R: The pearls of R are worth M-150, M-300, and so 
on, all the way down to M - 16*150, which is M-2400. So the total 
value of the right side is:

R = (M-150) + (M-300) + ... + (M-2400)
R = 16M - (150+300+...+2400)
R = 16M - 150(1+2+...+16)
R = 16M - 150*136
R = 16M - 20400

Okay! Now we just need to plug these values for L and R into our 
overall equation for the value of the necklace:

L+M+R = 65000
16M - 13600 + M + 16M - 20400 = 65000
33M - 34000 = 65000
33M = 99000
  M = 3000

So the middle pearl is worth $3000.  The pearls on the left side are 
worth $2900, $2800, down to $1400, while the ones on the right go in 
value from $2850, $2700, $2550, all the way down to $(3000-2400) = 
$600.  If you add these values together, you do indeed get the right 
answer (which is always nice!).

I hope this helps you with understanding this problem. Good luck!

-Doctor Daniel,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/   
    

Date: 01/14/03 at 15:51:05
From: Doctor Ian
Subject: Re: 33 Pearls Problem

Hi James,

Let's forget about the total number of pearls for a moment, and assume 
that we have one pearl, whose value is x.  Then let's add pairs of pearls,
following the formula given by the problem:

                 x 
               /   \
        (x-100)     (x-150)     <-- 2x -  250   
              |     |       
        (x-200)     (x-300)     <-- 2x -  500
              |     |      
        (x-300)     (x-450)     <-- 2x -  750
              |     |
        (x-400)     (x-600)     <-- 2x - 1000
           .           .
           .           .
        (x-100n)    (x-150n)    <-- 2x -  250n

Add them all up, and for (2n+1) pearls you get

   total value = x + n(2x) - 250[1 + 2 + ... + n]

               = (2n+1)x - 250[n(n+1)/2]

In this case, we have 33 pearls, so n=16; and the total value is 
$65,000; so 

  65,000 = (2*16 + 1)x - 250[16(17)/2]

         = 33x - 34,000

which gives a value of $3000 for the middle pearl.  

-Doctor Ian,  The Math Forum
 http://mathforum.org/dr.math/