There are three patterns below.
Two diagonal and one vertical. The middle number is common to them all What is it?
It is not mathematical per se.
NO partial answers. Explain all three patterns in your answer.
I think the middle number is
These are one more than the first five Roman numerals with one letter $(I, V, X, L, C)$.
These are the first five numbers which begin with a vowel in English.
When written in English the $n$th term of this sequence is the smallest which contains $n$ Es.
The quiz is very open to interpretation, so this is mine.
Every line sets a a lower bound for the numbers that can appear on the next. In the 1st line, the biggest number is 2, meaning that the second line contains only numbers > 2. The same logic applies in reverse, so 4th line smallest number is 17, hence, 3rd line must contain a number < 17.
Applying this rule, 8 > ? > 17. Suppose the aim of the game was to have all the digits in the highest number of each sequence (namely, 111 - 80 - 101) previously mentioned in the sequence itself.
111 can take the 1s from 1 and 17 but it's still missing one 1;
80 can take the 8 from either 8 or 18 but it's still missing the 0;
101 can take the 1 from 51 but it's missing both the 0 and one more 1.
This leads to the final solution: