1. Counting systems
Much loved at IOL (Eastern European tradition of “mathematical linguistics”)
Typically every other year (last time 2017)
Often involve some simple arithmetic 
And some semantics
Decimal system is very familiar, but you can be absolutely certain that, whatever it is, it won’t be base-10

2. Features of counting system
Most are base-10 for obvious reasons
But many have other bases, often mixed
Words may not be in the most logical order
Cf German etc dreiundvierzig 3+40 = 43
Often, a different base is used for counting different things
Look out for “counting back”
Cf Roman numerals IV, XC etc.
Or German clock time, “halb vier”, “dreiviertal acht”

3. Some familiar examples
English has special words for 11, 12, has one system for teens (U+T), another for the rest (T+U), but is quite regular
French is decimal until it gets to 70 (soixante-dix, soixante-et-onze, etc.) =60+10, 60+11, then 80 (quatre vingts ie 4x20), and 90 (quatre-vingt-dix ie 4x20+10)
Counting in 20s (“vigesimal”) is quite widespread,
even English has “four score and seven”

4. Example 1: Danish 1 en 2 to 3 tre 4 fire 5 fem 6 seks 7 syv 8 otte
10 ti
20 tyve
30 tredive
40 fyrre
50 halvtreds
60 tres
70 halvfjerds
80 firs
90 halvfems
23 treogtyve
47 syvogfyrre
71 enoghalvfjerds
?? seksoghalvtreds
95 ????
100 ????
100 hundrede 

5. Dozens “Dozenal” systems
Old English/Irish monetary system was 12- and 20-based
Makes calculation easier (divide by 2,3,4,5,6,8,10,12,16,20,25,etc)
In Nimbia (Nigeria) multiples of 12 are the basic number words around which everything else is built. The number 29 is gume bi ni biyar ((12x2)+5), and 95 is gume bo'o ni kwada ((12x7)+11).
This and other examples from

6. Example 2: Welsh (trad) 11 un ar ddeg 20 ugain 12 deuddeg
13 tri ar ddeg
14 pedwar ar ddeg
15 pymtheg
16 un ar bymtheg
17 dau ar bymtheg
18 deunaw
19 pedwar ar bymtheg
20 ugain
25 pump ar hugain
30 deg ar hugain
31 un ar ddeg ar hugain
40 deugain
41 deugain ac un
50 hanner cant
60 trigain
70 deg a thrigain

7. Example 3: Yoruba 1 ọ̀kan 2 èjì 3 ẹ̀ta 4 ẹ̀rin 5 àrún 10 ẹ̀wá 20 ogún
30 ọgbọ̀n
11 ọ̀kanlá
12 èjìlá
etc
15 aárúndíógún
16 ẹẹ́rìndílógún
17 eétàdílógún
18 eéjìdílógún
19 oókàndílógún
25 = 30-5
35 = 20x2-5
130 = 20x7-10
Some other complexities
Also, different words for counting objects

8. Weird bases Ndom (Papua New Guinea), has a base-6, number system.
It has basic words for 6, 18, and 36 and other numbers are built with reference to those. The number 25 is tondor abo mer abo sas (18+6+1), and 90 is nif thef abo tondor ((36x2)+18)
Huli (also PNG) uses a base-15 system.
Numbers which are multiples of 15 are simple words. Cf the English word for 225 and the Huli word ngui ngui, 15 15s. 80 in Huli is ngui dau, ngui waragane-gonaga duria ((15x5)+the 5th member of the 6th 15).

9. Mixed systems
In Bukiyip, another PNG language, there are two counting systems, depending on what you are counting.
Coconuts, days, and fish counted in base-3.
Betel nuts, bananas, and shields counted in base-4.
The word anauwip means 6 in the base-3 system and 24 in the base-4 system.

10. Oksapmin (NewGuinea)

11. How to solve counting system questions
Forget base-10, it will be something weird
They are often posed in the form of equations
Use your mathematical skills
Eg one question involved (some) sums of squares, but all number names were less than 10 – so you knew the words which were squared must be <4
Also, some morphophonemics maybe involved
Expect the unexpected

12. Writing systems Representation of spoken language Always derivative
All have pros and cons
Often borrowed from somewhere else
A few examples are purpose-built by linguists
Following typology a bit misleading: most writing systems are hybrid

13. Types of writing system
Pictographic
Elements are pictures, combined in graphically-interpretable patterns (e.g. temporal sequence or spatial relationship)
Ideographic
Elements denote ideas, combined in a logical fashion
Logographic
Elements denote words or morphemes, combined morphosyntactically
Syllabic
Elements denote syllables, combined phonologically

14. Types of writing system
Moraic
Like syllabic elements, but units are a bit smaller
Alphabetic
Elements denote phonemes (more or less), combined phonologically
Featural
Elements denote distinctive features of phonemes (such as voicing or place of articulation), combined phonologically
Also worth mentioning: direction?

15. eg 枘 枚 枝 枯 枰 柙 all names of trees (木)
Issues
Logographic
Typical example Chinese
Logographs typically contain a semantic element and a phonetic element
eg 枘 枚 枝 枯 枰 柙 all names of trees (木)
Some characters used phonetically (esp eg in transliterations of proper names), or as grammatical markers

16. Issues
Alphabetic
Often borrowed from another language, so compromises are necessary
Typical example: English uses Latin alphabet but has (many) sounds that Latin lacks, and so has to use letters inventively (sh, th, ch, wh, ee, ow, ...)
Very rarely truly phonetic, 1:1 correspondences
Same letter/sequence has different pronunciation
A given sound can be written in different ways

17. Brief digression
How logical (or not) is the English writing system?

18. Issues
Syllabic
May be systematic, eg Inuktitut

19. Issues
Syllabic
Or not, eg Japanese

20. Issues
Featural
Usually the result of linguistically sophisticated planning
Examples: Korean, Visible speech

21. How to solve writing system questions
Depends on what information you are given
Always question your assumptions, eg is it alphabetic, LtoR, etc.
Look for singletons
Look for high-frequency symbols or sequences
Look for “minimal pairs”
Bear in mind general “phonotactics”