# A Non Math Puzzle. What is the middle number?

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. I think the middle number is

$$11$$

First sequence

$$2,6,11,51,101$$
These are one more than the first five Roman numerals with one letter $$(I, V, X, L, C)$$.

Second sequence

$$1,8,11,18,80$$
These are the first five numbers which begin with a vowel in English.

Third sequence

$$1,3,11,17,111$$
When written in English the $$n$$th term of this sequence is the smallest which contains $$n$$ Es.

• Kudos @hexomino. Special for you. What are the next three numbers after 1,3,11,17,111 assuming 4-6 digit numbers are written as _ thousand etc. – DrD Aug 12 '19 at 15:26
• @DEEM I guess they would be 117, 317 and 1317. There's a little bit of a jump here but we can deduce from the previous terms in the sequence that there will be no term after 317 less than 1000 and adding 1000 just gives one extra E. – hexomino Aug 12 '19 at 15:36

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:

10