UnderTen by the numbers

UnderTen looks like a game of luck, but a deck of 52 cards follows rules of its own. Here is what the maths says about your starting hand, how fast hands shrink, and why a game of six feels so different from a game of two.

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Where these numbers come from

Some of the figures below are exact: they follow from counting the ways seven cards can be dealt from a 52-card deck. The rest come from simulations we ran with the same rules code the game itself uses. We played millions of rounds in which every player draws and throws the way the UnderTen bots do. Real people play differently from bots, so treat those figures as a guide rather than a law. We say which kind each number is.

Nobody can declare on the first turn

The lowest seven cards in the deck are the four Aces and three of the four 2s. Together they add up to 4 + 6 = 10. That is the smallest hand anyone can be dealt, and it is already one point too many to declare. So every round starts with at least one draw and one throw for each player, and the earliest anyone can declare is the start of their second turn. (Exact.)

What a starting hand is worth

An average card is worth 85 ÷ 13, about 6.5 points, because the 13 ranks add up to 1 + 2 + … + 9 plus four 10-point cards (the 10, Jack, Queen and King). Seven of them make an average starting hand of about 46 points. (Exact.)

In practice most hands land in the 40s:

Starting total	How often
10 to 29	about 2%
30 to 39	about 19%
40 to 49	about 46%
50 to 59	about 29%
60 to 70	about 4%

(Simulated over 400,000 deals.) The biggest possible hand is 70, which needs seven cards worth 10 each. There are 16 such cards in the deck, so a typical hand holds about two of them.

You will almost always have a 10-point card

Only about 6% of starting hands have no 10, Jack, Queen or King at all. In the other 94%, getting rid of those cards is your first job. (Exact: the number of 7-card hands drawn from the 36 lower cards, divided by all 7-card hands.)

Most hands start with a pair

About 79% of starting hands contain at least two cards of the same rank. Only 21% are seven different ranks. That is why sets matter so much: four times out of five, you can throw two cards in one turn from the very start. (Exact.)

How much deck is left

After the deal and the first face-up discard, the size of the deck depends on how many people are playing:

Players	Cards dealt	Left in the deck
2	14	37
3	21	30
4	28	23
5	35	16
6	42	9

At a table of six the deck runs out after a couple of trips around the table, and from then on it is made from reshuffled discards. Those discards are mostly the high cards people threw away, so a crowded table is a hard place to get low.

How long it takes to get under 10

This is the share of players whose hand has been 9 or less at the start of a turn by a given point in the round, in our bot simulations with nobody declaring:

Players	By turn 6	By turn 9	By turn 12
2	19%	68%	94%
3	20%	73%	97%
4	21%	72%	87%
5	18%	41%	60%
6	7%	24%	40%

(Simulated, 200,000 rounds per table size.) Two things stand out. First, almost nobody gets under 10 in the first five turns, so an early declare is rare and usually very strong. Second, at five or six players most hands never get there at all. That is why the game ends a round after 12 turns each: without the limit, a big table could go round and round.

Small tables and big tables play differently

• Two or three players. Plenty of fresh cards, so hands shrink steadily and most players get under 10 by their ninth turn. The question is not whether you can declare but whether you are lower than the others.
• Four players. The quick-match table. A good balance: most people reach a declarable hand, and timing matters most.
• Five or six players. Low cards are scarce and the deck recycles old discards. Being under 10 at all puts you ahead of most of the table, and the 12-turn limit often decides the round.

Our declare chart turns these numbers into a simple guide for when to call.

Why the 12-turn limit exists

The limit came out of the simulations we ran while designing the rules. As the table above shows, at a big table many hands never reach 9 or less, and without a limit those rounds would have no natural end. Twelve turns each is long enough that most players at a small table get a real chance to declare, and short enough that a six-player round stays a few minutes long.

Ready to test the numbers yourself? Play a round against the bots, or read the strategy guide first.