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Why might an auction go 1NT-3NT?
Sort of a follow up question to this question.
In that question, the auction goes 1NT - pass - 3NT - all pass. The answers to the question point out that, by opening 1NT, declarer is saying they have ~16 HCP. Their partner, bidding 3NT in response to 1NT, indicates they have ~12 HCP.
Since there are 40 HCP total, partner should be aware that they together control ~28 of HCP. Since the first opponent has already passed, this suggests that the last opponent is likely to pass as well (even if the last opponent has all 12 of the remaining HCP, the odds should be that they can't make any contract above 1NT with only 12 HCP). Therefore, partner should be able to conclude that opponents will only be passing for the rest of the bidding, i.e., the rest of the bidding will be uncontested.
Given that, why doesn't partner raise by a smaller amount, taking advantage of the bidding room to communicate their hand more clearly with declarer? It seems like they have a lot of bidding room to do so, and opponents won't be interfering anyway. Phrased alternatively, if a bidding system suggests that you should respond to 1NT with 3NT if you're holding ~12 HCP, then it seems to be giving up percentage points because it's not making use of the forecasted bidding room.
I'm wondering why a bidding system might recommend making this bid anyway, or if there's some flaw in the above analysis.

All of the intermediate bids between 1NT and 3NT have meanings. Trying to use them with a standard 3NT responding hand will dilute those other meanings, making it hard to find the right contract in other hands.

In principle you're right -- few pairs investigate 5-3 major fits when opener has a 5-card major but opens 1NT, and fewer investigate minor games, moysian (4-3) major fits, and staying in a partscore as potential alternatives to 3NT. Pragmatically, the edge you'd get if you could do these things will be negligible to negative because the alternatives are only very infrequently better than 3NT and even if you can bid what will be a very complicated system perfectly, you'll make opponents' defense easier.

I want to co-sign Tomek Czajka's answer and expand upon it a bit.
The reason the auction goes 1NT-3NT is that it maximizes your expected score. As a general rule, to make game in no trump with two balanced hands, you want a combined total of 25+ high card points (you can quibble one point in either direction). This isn't to say that you'll always make with 26 or always go down with 23. But: it is not efficient (or even really possible) to use bidding space to determine the exact composition of both hands to determine which suits are stopped, etc, and even if you did you're drawing a roadmap for the defense. Not all deals that go down double dummy go down in practice.
So: as responder after partner opens 1NT, if you are balanced (and don't think a major-suit game is worth exploring for), and you have values for game but not slam (so about 10-15 hcp), you should just bid 3NT. If you have a wide-open suit, the defense still has to find it with no real clues from the auction. It's also hard for them to know whether an active or passive defense is called for. As a defender against 1NT-3NT, holding KJxx xxx Qxx xxx, should I get active with a spade lead or stay passive with a heart or a club lead? It's impossible to say for sure, and if they guess wrong you collect the game bonus. And that's if they even have a right answer!

+1, the biggest reason to do this is to not tell opponents which (major) suit to lead after a failed Stayman

I think it would be good not to start by saying that it "maximizes the expected score", but focus on "gives minimal information to the defense".  There's no need for declarer and dummy to use the auction to exchange information about each other's hands that will be needed to plan the play, because no information could communicate to each other during the auction would tell them anything they wouldn't know as soon as dummy's hand hits the table.

@supercat but why do you care about giving minimal information to the defense? It's because it improves your expected score.

@ruds: That's a side-effect, that's also the reason for doing just about anything.  Merely knowing that the action improves the expected score doesn't clarify why.  Some bidding practices maximize the expected score by reducing the likelihood of the opponents landing a contract which is more favorable to them.  While going 1NT-3NT may reduce the likelihood that the opponents find a way to steal for less than the value of a 3NT contract (e.g. bidding 4S and going down only one), the primary value of its brevity is in the fact that it avoids revealing anything that could be used on defence.

@supercat No, it's not a side effect at all. The reason that you arrange your bidding system so that 1NT-3NT means what it does is because it maximizes your expected score. The mechanism by which it does that is by arriving at the contract that's likely to be the best while minimizing information to the defense. You can minimize information the defense by always passing -- if you always pass, you give away no information! Instead, you wish to maximize your score. In this case, it happens to be by arriving at a contract with minimal interaction because it doesn't provide too much info to opps.

@ruds: Many contracts resulting from 1NT-3NT that usually make in practice would yield a negative score in double-dummy play.  If in actual play, the contract resulting from 1NT-3NT would have a 60% chance of going +430 and a 40% chance of going -50, while a 4S contract would have a 100% chance of going +420, the contract resulting from 1NT-3NT would probably be optimal in practical play, but would be a lousy contract at double dummy.

Other bids are reserved for other hands. For example, if you respond 2NT, you're indicating 8-9 HCP rather than 12 HCP. By bidding 3NT directly, you're giving information that you don't have those other hands.
You're not giving up very much by ending the exchange of information in this situation because with a combined 27-29 HCP and no 8-card combined major suit, you're unlikely to find a better contract than 3NT.
Using other bids to exchange information in this situation would have more downsides than upsides:

it would prevent those other bids from being used in other situations when they are more useful
you'd be giving more information to the opponents

Agreed.  You're unlikely to find and diagnose a better contract, so just bid what you think you can probably make

The point count for the opening 1NT depends on the bidding system. In Standard American, it can be 15-17 or 16-18. In Acol, a weak no trump with 12-14 is common.
Assuming you're playing one of the above bidding systems, then the point count for the responding 3NT can be deduced by subtracting opener's point range from 25 high card points, which, as rude points out, is what you need on average to make game.
Why would you bid directly, rather than showing more of your hand? There are two major reasons, both involving the opponents. The first is to avoid divulging information. If you embark on an extended bidding sequence, arriving at 3NT having shown short diamonds, the opposition has an excellent view of where to attack. The direct sequence denies them this.
Secondly, if, over 1NT, you decide to show a suit, you make it easier for the opponents to come in. Once they have shown a suit, again, the opening lead is much easier. As explained above, the opponents may have up to 15 HCP, and if they have an unbalanced hand, two or three of their suit may be perfectly playable or an excellent sacrifice against your game.
One final point, which you did not address directly, but may be relevant for you: in competition bridge (pairs or teams), you bid games with a 40%-50% chance of making (depending on vulnerability). So you want to bid thin games, and if you can deny the opponents information, you will make more of them.

Even without shape for a sacrifice, extra bidding may let an opponent make a lead-directing double.

Because neither player is interested in anything else

Given that, why doesn't partner raise by a smaller amount, taking advantage of the bidding room to communicate their hand more clearly with declarer?

In most systems, 1NT is a relatively specific opening bid that gives much more information than a typical suit opening. Typically it indicates both a narrow HCP range and a balanced hand. If partner's hand is also balanced, it's between unlikely and impossible that a massive suit fit is being overlooked. On the rare occasions that, say, both players have a doubleton in the same suit, at least one of them can be expected to hold the A or K, so skillful play has a high probability of avoiding 5 losers. Overall, 3NT is likely to be made, and there is no reason to suspect 4H or 4S would be any easier.
Since 3NT is a game contract, there is no benefit to ending up at the 4 or 5 level. At contract it only moves a few points below the line, but you still would have made game anyway, and don't get any extra credit towards another game or towards the rubber. Similar logic applies to duplicate scoring.
Since excellent suit fits and shortages (i.e. ruffing potential) are both so unlikely, the only realistic route to slam at this point is through sheer high-card strength. But if you know that opener has no more than (typically) 18 HCP, it's easy to conclude that the strength needed for slam just isn't there.
Therefore, trying any other bids is at best a waste of time; at worst, it leaks information that could help the defenders select an opening lead or infer a weak or unstopped suit.

There are many, many ways to bid your hand after your partner opens 1NT. If you bid 3NT immediately, it shows (in most cases) a hand with no 4-card or 5-card major, that is strong enough for game but not strong enough for slam. So 3NT is very probably the best contract to play in.
With a 4-card major, you can bid 2 Clubs, the Stayman convention, that asks partner to bid their 4-card major if they have one. That way, you can reach four of a major if you have eight trumps between you.
With a 5-card major, you can make what is known as a transfer bid (if you have agreed this with your partner before the game). This shows five cards in the next suit up (so e.g. 2 Diamonds shows 5 Hearts); after partner bids 2 Hearts, you can make a further bid (2NT or 3NT with only five Hearts; 3 Hearts or 4 Hearts with six or more Hearts). So holding three or more Hearts, your partner will correct your NT bid to a Heart contract.
And so on.


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What card game is this
I used to play this 2 player card game with my nan when I was young, and was told it was called Chinese patience, but looking it up online I don't think it's correct, and I can't find anything like it. Any ideas?
You start by dealing all cards face down to both players, but as you are dealing you randomly put 4 cards face up along a row at the top of the table. The aim of the game is to get rid of all your cards, and the first to do so wins. You take it in turns, and on your turn you're trying to put cards on any of the 4 cards up top, usual solitaire rules apply there so if there is a red 7 it needs to be a black 6. If you have an ace, that goes to one side and that becomes an additional place you can get rid of cards, but in the usual sequence of needing the 2 next. If you can't go, you place your unused card face up next to your pile.
If the other player notices you missed something, you can then knock on the table, you then have to miss a turn, so you always need to pay attention! Another rule is that you can put cards on the opponents discarded pile if you have one that continues a sequence, so your 5 of hearts can go on their discarded 4 of hearts.
So that's about it. Anyone know what this might be called? Hoping my nan hasn't just completely made it up!
Thanks!

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Ecarte Bonus Point Scoring
In the card game Écarté, before play begins, if someone gets a bonus point after dealing that gets them to 5 points - do you still play that hand or does the game immediately end?

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Can a player play 'Sly Deal' again after 'Just Say No'?
Player A plays a 'Sly Deal' card,
Player B plays 'Just Say No',
Player A plays 'Sly Deal' again.
Is this valid? Will player B able to use the 'Sly Deal' card?

Your question is not clear. Is playing B playing a different Sly Deal card from their hand, or are they trying to play the same card they just played? Checking the rules It seems the first is fine and the second is not.