Earlier today I set you these six puzzles about succession with a lower case ‘s’. Here they are again with solutions.

**1. Nob job**

This is a masterpiece by the Japanese creator Nob Yogishahara, one of my favourite puzzles of all time.

What number goes in the circle with the ‘?’?

(No, the 7 in the final circle is not a mistake.)

**Solution**: 12

The rule is that you need to add the digits of the two numbers that point to the same circle. Thus, from top left, 7 + 2 + 9 + 9 = 27, and so on.

What’s great with this puzzle is that the obvious answer – that you take the difference between the two numbers pointing to the same circle – seems to work beautifully, until you get to the final circle, since 21 – 13 is not 7.

**2. Think Roman**

For this question and the following ones, what number comes next?

0, 0, 0, 0, 4, 9, 5, 1, 1, 0, 55, …

Hint: write out one, two, three, four, five, six, seven, eight, nine, ten, eleven, …

and look at the title of this puzzle.

**Solution: 55**

Roman numerals use the letters ‘i’, ‘v’, ‘x’, ‘l’, ‘c’ and ‘m’. Read only these letters, and treat them as if they are a Roman numeral. ‘Five’ is IV, ‘six’ is IX and so on. ‘Twelve’ is thus ‘‘LV’.

**3. Golomb’s sequence.**

1, 2, 2, 3, 3, 4, 4, 4, 5, 5, 5, 6, 6, 6, 6, 7, 7, 7, 7, 8, 8, 8, 8, 9, 9, 9, 9, …

Named after the eminent mathematician Solomon Golomb (whose work inspired the game Tetris), this is an extremely pleasing pattern once you spot it. Hint: think about why are there two 2s and two 3s.

**Solution:** 9.

The sequence follows this rule: The value of the number in position *n* is the number of appearances of the number *n*. Thus there is a single 1, two 2s, two 3s, three 4s, three 5s, four 6s…and five 9s.

**4. The elevator sequence**

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 14, 15, 16, 17, 18, 19, 20, 21, 22, …

Another puzzle is why this sequence got its name.

**Solution**: 23.

The sequence has every whole number except 13. Another description might be “Elevator numbers in many buildings” since in Western countries the absence of a 13th floor is widespread for superstitious reasons.

**5. The flagpole sequence**

What number comes next, i.e after THIRTEEN?

**Solution: **FOUR

It’s called the flagpole sequence for a reason! The flagpole (illustrated below) reads out the numbers in the list. So the next number must have a R in it since we can see that the next letter in the flagpole must be an R. Inspecting the previous numbers, we can see that numbers are never repeated, and the lowest acceptable number is always used. The lowest unused number with an R is four, which is the answer.

**6. Eban numbers**

2, 4, 6, 30, 32, 34, 36, 40, 42, 44, 46, 50, 52, 54, 56, 60, 62, 64, 66…

**Solution** 2,000

Again, the clue is in the name. If you ban all number words that include the letter ‘e’, you get this sequence…and rather wonderfully the successor to 66 is a long way down the line!

I hope you enjoyed the puzzles, I’ll be back in two weeks.

*Thanks to the Online Encyclopaedia of Integer Sequences and Eric Angelini for inspiring today’s puzzles.*

*I set a puzzle here every two weeks on a Monday. I’m always on the look-out for great puzzles. If you would like to suggest one, email me.*

*I give school talks about maths and puzzles (online and in person). If your school is interested please get in touch.*