The Network Effect

If you have the only cell phone in the world, it’s pretty useless, since you can’t call anyone. If there are 2 cell phones, there is one possible connection.  If there are 3 cell phones, you can make a total of 3 connections.  4 cell phones can have a total of 6 connections.  5 cell phones?  10 connections.   6 phones means 15 connections.

The more devices there are, the most connections you can make.  The more connections there are, the more useful the whole network becomes.  This is also called Metcalf’s Law.

Let’s look at the sequence of numbers generated above.

1, 3, 6, 10, 15, …

Can you see the pattern?   The number of connections can be represented by \frac{n(n-1)}{2}  where n is the number of nodes in the network.  Notice that this is very similar to \frac{n^2}{2} = \frac{1}{2}n^2.  So, the number of total possible connections is proportional to the square of the number of nodes in the network.  

Gimmicky Bidding Websites (Series/Sequence)

The way this site works is that if you bid, you have to pay that amount, even if you lose.  Bids are incremented by 1 cent.   Let’s say an item sells for 10 cents.  The guy who bid 1 cent still has to pay that, the guy who bid 2 cents still has to pay that, and so forth.   So, what does the auction site actually earn for selling that item for 10 cents?

Notice that 1+2+3+4+5+6+7+8+9+10  can be added up by grouping numbers from the opposite ends:  (1+10) + (2+9) + (3+8) + (4+7) + (5+6)   This is just 11+11+11+11+11  or 11*5 = 55.  Note that when x = 10, and we ended up multiplying 11*5 for the series sum.

So, the general formula is:

1 + 2 + 3 + … + n = \displaystyle\sum\limits_{x=0}^n x = (n+1)\frac{n}{2} = \frac{n^2+n}{2} = \frac{n(n+1)}{2}

Pop Quiz!  If the sunglasses in the photo end up selling for $6.96, how much does the website make?  \frac{696*697}{2} = \frac{485112}{2} = 242556 = \$2,425.56   !!