Sums of Products of Similar Numbers Across Sets Get Bigger Results
In Box C the second set of numbers is reversed so that the number at the opposite ends of the range are multiplied. The first pair is 1 and 5, and the last pair is 5 and 1. The sum of the products results in the smallest possible number. In Box B where the pairs were the most similar the result was the largest possible number and in Box C the pairs were the most different the smallest number resulted. Multiplying pairs and summing the results tells us something about the arrangement. You get the smallest result when you multiply the smallest and the largest with their opposite.
In Box D the second set of numbers has been changed around a little bit so that the summed products of pairs is somewhere in between the largest possibility and the smallest possibility. That is, since the two smallest are together the next two biggest are together. That indicates that small things are going together with small things and large things with large things. So we say that's a relationship between those two sets of numbers then.
Intuitive understanding of how prototypes #2 and #3 indicate whether or not two sets of numbers are similar (related). The principle underlying the combination of these two prototypes is that when two sets of numbers are paired together so that the numbers of each set are paired with their most similar size in the other set the resulting sum products will be larger than if they are not paired with their most similar size. The purpose is to get an intuitive grasp of this principle. Only two sets of two numbers each are used.
This first example shows the most similar pairs together (the 2 goes with the other 2 and the 5 goes with the other 5).
5 X 5 = 2 5
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29
This next example shows the dissimilar pairs of the 2 of the first set goes with the 5 of second set and the 5 of the first set goes with the 2 of the second set.
2 X 5 = 10
5 X 2 = 10
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20
Notice in the first set that in the 5 X 5 pair that there are 5 rows of 5-one more row of 5s than there are in all pairs of the second set where there are only 4 rows of 5s combined. That is, when bigger numbers are multiplied together they produce even bigger results. When the larger numbers are paired with larger numbers then the resulting products with be larger than if they were not paired together. This occurs even when the two sets of numbers are on different scales (the numbers in each set do not need to be the same). Two more sets of two numbers show the principle.
This example shows the smallest pair of the set together (2 and 6) and the largest pair of the set together (4 and 8).
2 X 6 = 12
4 X 8 = 32
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44
The next example shows the smallest number of the first set paired with the largest number of the second set paired together (2 and 8) and the largest number of the first set paired with the smallest number of the second set (4 and 6).
2 X 8 = 16
4 X 6 = 24
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40
This example is not as dramatic as the first in showing the bigger result of big numbers but it still exists. Notice in the first example in the larger pair (4 and 8) there are 4 rows of 8s compared to the second example where there are 4 rows of 6s (a loss of 8). At the same time when the 2 is multiplied by the 8 there is only a gain of 4 over the 2 times the 6 in the first set. The principle holds that when pairs numbers of similar size are multiplied of two sets the sum of the products will be larger than any other possible pairing.
