21X
21X Rulebook
Table of Contents
Table of Contents
Components
110 Cards | 6 Help cards
Note: you will need a 1 minute timer (or a stopwatch for single player).
Goal
The goal of 21X is to make 21 without going 'bust', just like the traditional card games Blackjack, 21 and Pontoon. But 21X has a twist: the cards are algebraic expressions!
Players race to find a value for X that makes their cards add up to 21. The first player to make 21 or be the closest wins. Any total higher than 21 is 'bust' – an invalid total that cannot win the game.
Set-Up
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Decide which cards to play with. In the corner of each card there are icons indicating the card's difficulty level. The frame of the card indicates which set it is from:
- Level 1: one triangle
- Level 2: a square and a triangle
- Level 3: a pentagon, a square and a triangle
Sets:
- White corners – Core
- Black corners – Exclusive Limited Edition
- Golden borders – Celebrity
If this is your first game, we recommend you only use the level 1 core set cards. If you're confident in maths, you can add more cards! We recommend using pen and paper for your first game if you have it.
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Shuffle the chosen set(s) to form a deck.
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Deal 2 cards to each player. Place the deck face down in the centre of the table.
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Place the help cards within easy reach of all players.
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All players simultaneously reveal their cards and start playing. Do not start the timer!
The Cards
Most cards have N or X on them.
The value of N is equal to the number of cards that player has. e.g. for a player with 3 cards, N = 3.
You choose the value of X following these limitations:
- X must be the same across all your cards.
- X must be an integer (whole number) but can be positive or negative.
- X cannot be 0.
How to Play
Players play simultaneously with their cards face up. There are no turns.
At any time, players may do any of the following:
Draw: you draw the top card of the deck and keep playing. Players cannot have more than 5 cards each. Remember that when you draw, your N value increases as well.
Call 21: all players stop immediately. State your X value and sum your cards clearly so that other players can check your work. If you made 21, you win and the game ends. If you did not, you are eliminated and play continues (you can join in again next game!).
Stand: state your final total and your X value. Start a 1 minute timer. Your total and X value cannot be changed anymore. Other players keep playing, trying to beat your total. They may draw and call 21 but cannot stand.
Note: a total of exactly 20 is also 'bust' (see FAQs on p6).
At the end of the minute, all players stop immediately.
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All players state the best (highest valid) total they were able to make before the timer ran out. Any players hesitating or still playing are eliminated.
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Check the X value and maths of the player with the best total (or the standing player if nobody beat their total). If they are correct, they win. If not, check the work of the player with the next best total and so on until you find a winner.
If you are only using the core cards you can play now!
Example
This player has drawn once and now has a 3 card hand. Here are their calculations for 3 different values for X. The hand is the cards 6X, 2X+4 and X/2.
- X = 4: 6x4 = 24; 2x4+4 = 12; 4/2 = 2. Total: 24+12+2 = 38. Bust!
- X = 1: 6x1 = 6; 2x1+4 = 6; 1/2 = 0.5. Total: 6+6+0.5 = 12.5. Could stand but unlikely to win.
- X = 2: 6x2 = 12; 2x2+4 = 8; 2/2 = 1. Total: 12+8+1 = 21. Call 21 and win the game!
Celebrity Cards
These are golden with detailed graphics.
These cards all function the same. Take your X value and count that many places along the sequence on the card. The card's value is equal to the number in that position (in-depth descriptions on page 6).
Note: if your X value is negative, process the card as if X were positive, then make the result negative. This isn't mathematically accurate... but it makes these exciting cards function!
Exclusive Limited Edition Cards
These have black corners.
They add V and Y to the game.
The value of V is equal to 1+ the number of games of 21X you have won this session. e.g. if you have won 3 games, V = 4.
You choose the value of Y. It follows all the same rules as X with one exception: Y must be positive.
Difficulty
The game has three difficulty levels for adapting the game to the players' maths knowledge. Here is a further explanation of the skills required:
- Level 1 (one triangle): Integers and simple expressions. Negative numbers. Addition, subtraction and multiplication.
- Level 2 (a square and a triangle): Divide by 2, brackets, indices/powers (squared, cubed and so on).
- Level 3 (a pentagon, a square and a triangle): More complex division.
- Celebrity cards: famous number sequences, irrational numbers, factors, primes etc.
- Limited edition cards: factorials.
Other difficulty options:
- Allow pen/paper (reduces difficulty).
- Allow calculators (reduces difficulty).
- Allow non-integer values for X (increases difficulty).
- Allow players to have more than 5 cards (increases difficulty).
Scoring
To play competitively, you can keep track of each player's score:
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Choose the number of games you want to play (optional)
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At the end of each game, the winner scores points equal to the sum of difficulty icons across all their cards. Record this, then proceed to the next game.
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When you have played the agreed number of games (or you collectively decide to stop), whoever has the most points wins.
Single Player Rules
Objective
Score 21 points (a sum of 21 difficulty icons – see 'scoring' above) as quickly as possible by making hands that total exactly 21.
Set-up
Set-up is the same as multiplayer with two exceptions:
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When choosing cards to play with, you must include all the core set cards. Use as many or as few other cards as you like.
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When you start playing, start a stopwatch.
How to play
At any time you may do any of the following:
Draw – same rules as multiplayer.
Mulligan – discard your entire hand then deal yourself 2 new cards and keep playing. You can do this as often as you like.
Make 21 – this has 3 steps:
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Choose an X value and check your work. If you made 21, set your hand neatly aside in a score area (keep scored hands separate from each other). If not, skip steps 2 and 3 and keep playing.
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Check if you have ended the single player game (see below).
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If the game did not end, draw 2 cards to start a new hand and keep playing.
Other rules:
- In single player, V is equal to 1+ the number of hands currently in your score area. All other cards function identically to multiplayer.
- You cannot stand while playing single player.
- If you need to draw more cards but the deck is empty, shuffle the discarded cards to form a new deck, then draw from there.
Single player game end
You win when you have 21 points (a sum of 21 difficulty icons) or more in your score area. Stop the stopwatch and record your time. See if you can beat this result next time!
Celebrity Cards in Detail
Xth prime by Marcus du Sautoy PhD
Start in the centre and work outward around the spiral. If X = 5, this card is worth 11. If X = -7, this card is worth -17. Primes are positive integers greater than 1 which can only be divided by 1 and themselves.
Xth triangle by James Grime PhD
Start at the top of the triangle and work downward. If X = 4, this card is worth 10. If X = -2, this card is worth -3. Triangle numbers are the sum of all integers up to X. e.g. 1+2 = 3, 1+2+3 = 6 and so on.
Fib (X) (aka Fibonacci sequence) by Ben Sparks
Start in the top left and count along, moving down to the second row if needed. If X = 4, this card is worth 3. If X = -9, this card is worth -34. Fibonacci numbers are the sum of the two previous numbers in the sequence. e.g. 2+3 = 5, 3+5 = 8 and so on.
Sum of factors (X) by Katie Steckles PhD
Simply count from left to right. If X = 6, this card is worth 12. If X = -10, this card is worth -18. This sequence sums all the integers X can be divided by. e.g. If X = 4, 4 can be divided by 1,2 and 4. 1+2+4 = 7. If X = 10, 10 can be divided by 1,2,5 and 10. 1+2+5+10 = 18.
Xth digit of pi by Matt Parker
Starting with the 1 after the decimal point, count clockwise around the circle. If X = 4, this card is worth 5. If X = -19, this card is worth -4.
Xth digit of e (aka Euler's number) by Tom Crawford PhD
Starting with the 7 after the decimal point, count from left to right. If X = 1, this card is worth 7. If X = -8, this card is worth -2.
The "Xth digit of" cards provide a digit from a famous irrational number. They're more chaotic to work with!
21X FAQs
Can I get rid of cards?
No. Once you have drawn a card, you must keep it.
Can I change X values in my head?
Yes. Your X value is only final when you call 21 or stand. Until then, you may change X freely. You cannot go 'bust' in your head.
How are ties decided?
The standing player wins ties. If non-standing players tie, the player with the most difficulty icons wins. If still tied, all tied players win.
What happens if I call 21 during the 1 minute timer but get my maths wrong?
You are eliminated, then you restart the timer.
Can I stand/win if my total is a non-integer?
Yes. X must be an integer but your total does not.
Why is a total of exactly 20 bust?
It makes the game too easy. Most cards are even and you start the game with an even number of cards. Because of this, the chance you can make exactly 20 is very high compared to other totals.