I think I can explain it to you in a more didactic way.
The final scoring "two-digit" total in tenhou is the result of many rules that are agreed upon (or understood) before playing.
Understanding the rules/bonuses
Even though all players usually start a mahjong round with 25,000 points, sometimes they "owe" more than that, usually 30,000 points (virtually, of course, you never see this until you calculate the final scores). This is the case in tenhou. Think of it as a "buy-in", like in poker. Though all players have 25,000 points at gamestart, they have actually "paid" the equivalent of 30,000 to get into the game. This also implies that all players don't start the game at +-0 points, but that they all start with an implied score of -5, further encouraging the chasing of points.
When players start the game holding 25,000 points with a buy-in of 30,000 points, it will appear in the rules as 25000点持ち、30000点返し
nimangosenten-mochi, sanmanten-gaeshi or "25k to go, 30k to come back".
In many cases, as it is the case with tenhou, these missing 5000 points from all players are to be awarded to the winner of the game. Like when you build a pot in poker. Therefore, it is said that the winner of the game will receive 20.000 extra points at endgame (5.000 points x 4 players). This award is usually called オカ (oka) and in few cases "1st place award" (but do note there are many more types of first place awards).
- Also, do note that these rules generally imply playing with a 西入 or 南入 (shaa-nyuu or nan-nyuu) setting, where the game will not end after the last hand in case a player hasn't gotten over the 30,000 points mark. Tenhou implements this, along with a "sudden death" (サドンデス) rule, meaning that at the time a player gets past the 30,000 points mark in the extra round, the game will end. Furthermore, in tenhou, no second extra round is ever played, meaning that a full game will never go into North round, and a half game will never go into West round.
Along with this 20K bonus for first place, additional bonuses are implemented to encourage competition for relative placing. Usually, the top two positions receive points, and the last two positions pay points. This is called Uma (馬, horse) and it is fairly common to find many types. The most usual distribution is 1-2 (ワンツー) which implies a +20K +10K -10K -20K points symmetrical point distribution among the 1st, 2nd, 3rd and 4th place players accordingly. The sum of uma bonuses should, in most cases, equal to 0. Tenhou uses this 1-2 distribution.
Calculating the score
Take the raw endgame scores.
39200
28100
20700
12000
Pull three decimals to the left (i.e. divide by a thousand).
39.2
28.1
20.7
12
Now, this is where there are different takes on how to proceed. Very few places leave the final decimal just as it is, and this assures more precision but requires more space on paper, more time inputting data, and more data to manage altogether, increasing the nuisance of managing point databases (both virtual and otherwise). The usual thing to do is "round" these numbers to the closest integer (i.e. a number without points). What about .5? There are two takes: 1. 四捨五入 "rounding
up" and 2. 五捨六入 "rounding
down." Tenhou, and most places, apply the former system, so you will round up any .5 or more you find.
39
28
21
12
Only 20.7 was rounded up. Now, you may ask, "but, won't rounding up make the final scores not add up to 0!?!" Whenever this happens, the difference is taken from the first place player. And this is why, when calculating final scores manually,
you don't calculate the points for first place to make things faster. Instead, the final scoring for first place will be the sum of all three other scores. I will still include the first place scoring in my examples so you can see the progress better, but
keep that in mind.
Next, you remove the 30 (thousand) points from the "buy-in" (or whatever you want to call it).
+9
△2
△9
△18
Note that minuses are usually drawn as triangles, as they are easy to mistake with + signs if the sheet gets some dirt or ink remains on it.
Finally, you add bonuses. As I said, we don't usually take the first place player into account, so when ignoring him, you skip the oka bonus part and go to adding/substracting uma immediately. As this is a visual example to show you go final scores are obtained, I will include this calculation:
Oka
You add the 20 (thousand) points "pot" (5 thousand points x 4 players) to the first place player.
+29
△2
△9
△18
You add/substract uma accordingly. In the case of tenhou, +20 +10 -10 -20
+49
+8
△19
△38
And finally, you sum the scores of the players in 2nd, 3rd and 4th place to get the score for the player in first place. Sometimes it will match the calculations, but sometimes there will be a small difference (1 or so).
+8 -19 -38 = -49
Remember to invert the sign. So, the final score for first place is 49. In this case, it matched that one in the calculation, but remember to always do this last sum as sometimes it doesn't. When you do this, it explains the final score you get at the end of the game in tenhou:
+49
+8
△19
△38
I hope this helped you understand why and how this is so, and how the "scoring rules" work and relate to each other.
EDIT:
I am gonna add a cheat-sheet for you when using tenhou-like scoring rules.
Shortcuts
Don't subtract or add anything. Just use this cheat formula. Note that irregularities and mistakes may and will often occur in the final scoring due to not following the natural process. Basically, all you are doing is adding up and canceling bonuses.
- 1st place +10 (or just ignore and sum the rest)
2nd place -20
3rd place -40
4th place -50
+-0
The amount of points necessary for each placement to achieve a +-0 score.
1st place: N/A (impossible when first place)
2nd place: from 19.500 to 20.400 points
3rd place: from 39.500 to 40.400 points
4th place: N/A (impossible when fourth place)
--
EDIT2:
Further reading!
Your other question
The points that Jenn and Garthe refer to in the podcasts when they say they ended with "x points that session/day" is the sum of their final scores after a number of games which I believe is 4 hanchan. Furthermore, the league season has several sessions.
More
Even at Jenn and Garthe's league, different bonuses and scoring systems are applied from those you would be used to. Scoring rules are, perhaps, the part of mahjong most likely to be due to change in different places.
At my local mahjong club, I use 30,000 starting points, no oka, and a 1-2 uma. This is due to many facts. While I want to still encourage placement in-fighting with uma, I don't want to add any further extra emphasis or weight on ending in 1st place. Furthermore, since we play with tobi (bankruptcy rule) a more elevated starting point number is desired to limit the amount of games ending earlier than expected. In other words, the scoring rules you will use depends on what approach you want or need the players to take and, analogically, players will decide how to play more efficiently according to the scoring rules being used.
The reality is, that scoring in tenhou's ranked games is useless, as you are not playing for scoring but for placement. And, in a wider definition, for a placement that is not 4th. Never will they affect your rank or rating, however they may affect your total placing in the monthly rankings (which is an useless and ever shifting bragging rights system for whoever is on first at the time, especially in higher levels).
Exchanging money for cash
A value is assigned to every of these points before playing. You can usually find .1 (ten ichi) which means "10 yen per point" or .2 (ten ni) which means "20 yen per point" at home-games; .3 (ten san) which means "30 yen per point" at some low-rate parlors; .5 (ten go) which means "50 yen per point" at most parlors and 1 (pin or ten pin) which means "100 yen per point" in every commercial parlor and is currently the most popular among pseudo "gamblers" and middle aged men, and is widely accepted to be the highest "legal" rate for parlors, though I have seen a 2 parlor (twice that value) once. Any higher than that would be considered gambling and is illegal, though it is not impossible to play it. There are sort of "underground parlors" for people who know them, usually in the form of "mansion mahjong" or "playing in a mansion" where the rate can go as high as to the highest played rate I know, 10 (deka pin) which pays 1000 yen per point (this means that your starting 25000 point are virtually 25000 yen. Scary). You don't need to be a rocket scientist to find out how much money your points convert to for each system (multiply your points for the rate).
Also, do note that after converting the points to money, free parlors will take a little bit from each player to collect the "table rake". In some cases, they may take an extra amount from the player in first.
On top of that, if the parlor uses "chips" (shuugi) to award certain events, and you really want to keep count of how much you are winning/losing every day, you need to either count the money that you have put in for parlor, or calculate the money value of those chips that you have given/received. Usually, it is 100 yen per shuugi in tengo and 500 yen per shuugi in tenpin. Tenhou has a special rooms for different values for those shuugi in its Jansou tab.
Finally
If it took you 30 games to get one first place, you might want to check this blog out
http://www.osamuko.com/category/theory/
