# What is the rule these numbers follow?

Each row in this grid of numbers follows a specific pattern or rule:

 7   1   5   2  ||   8   4
12  10  14  13  ||  12   6
11   2  14   2  ||  15   6
8   3  19   3  ||  21   6
6   3   8   2  ||  16   3
10   3  19   3  ||  21   7
11   5  26   4  ||  28   9
8   6  11   9  ||  22  19


There are two goals: Firstly, you must determine what the pattern is, and secondly, you must create a row of your own that follows the pattern.

Hint 1:

There is a reason that every second number in a row is smaller than the number that comes before it.

Hint 2:

The seperator is there for a purpose also

Hint 3:

This puzzle works regardless of base or representation

Hint 4 (major hint):

Half of the numbers, when removed can be uniquely replaced, while the other half each could potentially be replaced with one other number.

• Are the numbers all unique? As in, if I took away any one of them, would a person 'in the know' put back that number and can put back only that number? – eedrah Oct 24 '17 at 2:08
• see hint 4 for the answer to that – micsthepick Dec 13 '17 at 21:07

## 1 Answer

The pattern is that

If you treat pairs of numbers as binomial coefficients, multiplying the two coefficients on the left gives the value of the coefficient on the right.

Examples:

$$\binom{7}{1} \times \binom{5}{2} = \binom{8}{4}$$
$$\binom{12}{10} \times \binom{14}{13} = \binom{12}{6}$$
$$\binom{11}{2} \times \binom{14}{2} = \binom{15}{6}$$
$$\binom{6}{3} \times \binom{8}{2} = \binom{16}{3}$$
$$\binom{10}{3} \times \binom{19}{3} = \binom{21}{7}$$
$$\binom{11}{5} \times \binom{26}{4} = \binom{28}{9}$$
$$\binom{8}{6} \times \binom{11}{9} = \binom{22}{19}$$

Example of my own finding:

8 6 6 3 || 16 3

• I'm not surprised that you figured this out, ffao, but it would be nice to see some mention of what made the pieces (maximum prime factors?) fall into place – humn Oct 24 '17 at 17:14