Friedrich Gauss (1777-1855), the German mathematician (considered to be one of the three greatest mathematicians in history, the other two are Archimedes and Albert Einstein) was best known for the Gaussian bell curve, such as in high school, a grade-histogram or distribution for correcting grades based on relative “fairness enforcement”, independent of students outside current class.

Usually, the A’s and F’s are less common, off to two opposite sides, but the C’s more common in the middle. Gaussian statistics became the time-transcending main-stay for most forms of statistical analysis since the 1800’s.

Gauss’s first great achievement as a child was quickly adding up all the numbers from 1 to 100, and was ridiculed by his teacher for it at first, expecting him to take many minutes like the other students. He quickly accessed a “pattern” (such as the pattern-matching we do here with our scrapers) where 1 + 2 + 3 + … + 100 means (1+100) + (2+99) + (3 + 98) + … + (50 + 51), (each addition pair = 101) which, because it’s an even number, and pairing numbers means 50 possible additions, 50 being half of 100, means you multiply 101 by 50 = 5050. If the final number is odd, just start with 0 to make the number of numbers even, and it results in the same formula, **n(n + 1)/2**, where n is the highest number, 100 for the above example, see below for more detailed explanation.

**The point I am making is that you should reward each affiliate below the above affiliate, cascading downward, with a slightly lower percentage commission to motivate the higher ranking affiliates.** Your 99-tier affiliate/reseller pyramid is no different from the Gauss problem-solution analysis above, only you’re adding 1 to 99, and you want each commission awarded (1% instead of “1”) to always exceed the 30-cent transaction fee charged by PayPal + 2.9%, or 33 cents or higher. Physical goods usually have a lower profit margin, especially when you’re a drop-shipper, but will go up when pooled-whole-sellers buy in bulk. Ideally, you want to pay no less than 5% for the first-tier affiliate, but the highest tier will have more acceptance of lower commission percentages, because they will be backlinked by you and receive credit for all sales from surfers sent to you by affiliates below them. **But always remind them the real goal is for them to rank higher on Google/search engines, the seven advantages of automatic outbound links, and if they receive a lower percentage commission, your profit margin is higher, to justify more profit sharing/bonus-commissions if you choose to redistribute. **Digital goods (eg. software, movies, music), because they are instantly reprinted with little-to-no added costs when purchased in volume, generally dictates a minimum percentage of 20%. But if you add 1 to 5 or 1 to 20, that’s not 99 transitions, so should you use fractions/decimals? How does this change the formula?

Use the example of 1 to 5. Rounding 99 up to 100, 100 divided by 5 is 20. So each number should be broken into 0.05 steps (1.0% + 1.05% + … + 5.0%). For 1 to 20, 100 divided by 20 is 5, 0.2 steps, so (1.0% + 1.5% + … + 20%). Adding these numbers as decimals uses the above algorithm, (1.0% + 5.0%) + (1.05% + 4.95%) + …, only now there are 20 times as many additions, so **n(n + 1)/2** is multiplied by 20, and 5 for 1 to 20, with 0.2 steps. The answer: 300% for n=5, .05 steps, and 1050% for n=20, 0.2 steps. That means, in the very rare but worst-case scenario event, someone sent from the lowest level affiliate and makes a purchase, you lose 3 to 10.05 times what your affiliates earn for you. But because on average the surfer comes from much higher ranking affiliates (not at the midpoint, or the 49-50th tier, but higher because on average 90% of their traffic comes from search engines, greater percentages of the pool of affiliates when they rank higher), **combined with that fact you are in a good position to be #1 on that keyword and will probably keep it that way with their invitations for more related trending keywords in anchor text for new promotions. **You get exponentially higher numbers of search engine visitors.

Here is a comparison of the Click Through Rate for the top organic results for each study (more information at https://moz.com/blog/google-organic-click-through-rates-in-2014):

Working at CPA revenue-share advertising firms in the past revealed the successful promoters, even though CPA is designed to be no-risk (but with a “creative” fee) will gamble with potential initial losses, knowing they are competing against other promoters, even when they sell completely different products/services on similar ad services, commercial breaks, etc. A “buffer”, or projected potential money-loss with money in bank, is wise, but your risks are much lower here. Just be advised that organic indexing (where you get credit for backlinks) occurs 6 weeks after backlink/HTML is indexed by Googlebot, but you don’t pay your affiliates/resellers until 2-6 weeks after the first possible commission you have to reward them with.

**Techniques for Adding the Numbers 1 to 100**

There’s a popular story that Gauss, mathematician extraordinaire, had a lazy teacher. The so-called educator wanted to keep the kids busy so he could take a nap; he asked the class to add the numbers 1 to 100.

Gauss approached with his answer: 5050. So soon? The teacher suspected a cheat, but no. Manual addition was for suckers, and Gauss found a formula to sidestep the problem:

Let’s share a few explanations of this result and really understand it intuitively. For these examples we’ll add 1 to 10, and then see how it applies for 1 to 100 (or 1 to any number).

**Technique 1: Pair Numbers**

Pairing numbers is a common approach to this problem. Instead of writing all the numbers in a single column, let’s wrap the numbers around, like this:

1 2 3 4 5

10 9 8 7 6

An interesting pattern emerges: **the sum of each column is 11**. As the top row increases, the bottom row decreases, so the sum stays the same.

Because 1 is paired with 10 (our n), we can say that each column has (n+1). And how many pairs do we have? Well, we have 2 equal rows, we must have n/2 pairs.

which is the formula above.

**Wait — what about an odd number of items?**

Ah, I’m glad you brought it up. What if we are adding up the numbers 1 to 9? We don’t have an even number of items to pair up. Many explanations will just give the explanation above and leave it at that. I won’t.

Let’s add the numbers 1 to 9, but instead of starting from 1, let’s count from 0 instead:

0 1 2 3 4

9 8 7 6 5

By counting from 0, we get an “extra item” (10 in total) so we can have an even number of rows. However, our formula will look a bit different.

Notice that each column has a sum of n (not n+1, like before), since 0 and 9 are grouped. And instead of having exactly n items in 2 rows (for n/2 pairs total), we have n + 1 items in 2 rows (for (n + 1)/2 pairs total). If you plug these numbers in you get:

which is the same formula as before. It always bugged me that the same formula worked for both odd and even numbers – won’t you get a fraction? Yep, you get the same formula, but for different reasons.