## Patterns

The solution to some problems may depend upon one's being able to discover a pattern in an array of numbers or values. Frequently, it is convenient to examine the patterns in a sample rather than the entire population. Once a pattern is discovered and shown to be consistent for the sample, it can be used to predict the solution for the entire population.

Table 1.1. Patterns in numbers, first sample.

Cow number

Table 1.1. Patterns in numbers, first sample.

Cow number

 Ration Child 1 2 3 4 5 6 7 8 9 10 Grain 1 N N N N N N N N N N Mineral 2 Y Y Y Y Y Hay 3 Y Y Y Silage 4 Y Y Water 5 Y Y

Problem: A dairy farmer has five children. Each child is responsible for one part of the daily feed ration for the family's 100 dairy cows. The oldest is responsible for the grain, the second for the minerals, the third for the hay, the fourth for the silage, and the fifth for water. Instead of feeding each cow, the first child decides she will not feed the cows at all that day. The second child decides just to feed every other cow, the third child feeds every third cow, and so on. Dad soon discovers how the cows were fed, and needs to know which cows did not receive any feed or water.

Solution: When one is faced with this type of problem, it is usually helpful to set up a table. In this case, it would be very time-consuming to set up a table for all 100 cows. Instead, select a sample of the cows. If a pattern is true for the sample, there is a high probability that the pattern will be true for a large group. Determining the size of a sample is not always easy. Pick one, and if a clear pattern does not appear, increase the size until a pattern develops. We will start with the first 10 cows, Table 1.1.

In this sample, cows #1 and #7 did not receive any grain, mineral, hay, silage, or water. Is this enough information to establish a pattern? We will predict that the next cow that did not receive any feed or water is #11. Why? To test this prediction, the sample size must be extended to include a larger number of cows.

Table 1.2 shows that the prediction was right; cow #11, along with #13, #17, and #19, did not receive any grain, minerals, hay, silage, or water. It is now safe to consider that the prediction could be used to identify all of the animals within the herd that did not receive any grain, minerals, hay, silage, or water (those animals represented by prime numbers, that is, a number divisible only by itself and one).

Table 1.2. Patterns in numbers, second sample.

Cow number

Table 1.2. Patterns in numbers, second sample.

Cow number

 Ration Child 11 12 13 14 15 16 17 18 19 20 Grain 1 N N N N N N N N N N Mineral 2 Y Y Y Y Y Hay 3 Y Y Y Silage 4 Y Y Y Water 5 Y Y