Four Fours
This is a competitive version of the classic arithmetic problem of expressing the numbers from 1 through 20 using exactly four 4s and the standard arithmetic operations.
Equipment
Send $20 to me for game equipment. Or you can make your own using a set of 100 3 by 5 index cards. Your cards should include: 20 + signs, 20 - signs, 20 × signs, 20 ÷ signs, and 20 (, i.e., 20 "open parenthesis" symbols. Note that the ( cards can be turned over to be a ).
You will also need a round tracker (see below) and a game board consisting of four 4s, with enough space between them to play a 3 x 5 card. If you're making your own set, note that there is one game board shared by all players.
Play
The game is played in 9 rounds. The goal of each round is to express the round number as a complete arithmetical expression using the four 4s and the standard mathematical operations. For example, if the round number is 8, and a player completes the expression (4 + 4) × (4 ÷ 4), that player wins the round.
Shuffle the cards, and deal 5 cards to each player. Set the round number to "1." Each player can lay down up to three cards on the game board. These cards can be played:
In an empty space (if any) between 4s,
On top of a different card already on the board.
You may not play a card on top of a similar card (for example, you may not play a + on top of an existing +), and you may not play a card on top of a card you have just played.
Parenthesis cards can be played either an open parenthesis ( or a close parenthesis ). They are subject to the following rules:
A parenthesis placed before the first 4 must be an "open" parenthesis: (.
A parenthesis placed after the last 4 must be a "close" parenthesis: ).
You may play a parenthesis on top of another parenthesis. If this occurs, both parentheses are removed from the board. (This is the only way to eliminate a parenthesis from the game board)
You may play a parenthesis before another parenthesis of the same type. For example, you can play a ( before another (. However, there is a limit of three consecutive parentheses. Thus the expression 4 + ( ( ( 4 is allowable; however, you may not place a fourth parenthesis before the second 4.
There is no implied multiplication. For example, if you place a parenthesis between two 4s to form 4 ( 4, an operation must be placed between the first 4 and the ( to complete the expression. However, see the optional rules.
Parentheses do not have to be matched when played. For example, it is permissible to leave the board showing an expression like 4 + (4 - 4 ( 4.
The player then draws cards until they have 5 in their hand again.
If the expression on the board is incomplete, either because the parentheses are not matched or there are insufficient operands, play passes to the person on the right.
If the expression on the game board does not equal the round number, play passes to the person on the current player's right, who continues building an expression on the game board.
If the expression on the board is complete and equal to the round number, the current player wins the round and scores a number of points equal to the total number of cards on the board (including cards beneath other cards). The round number is advanced to the next number and all cards are removed from the board. All players, beginning with the player to the current player's right, may discard up to three cards from their hand and redraw so they have five cards in their hand again. Play then passes to the person on the current player's right.
The game ends when the 9th round is complete. However, see optional rules.
Example: In Round 1, the first player sets down a × and + card to obtain the expression 4 × 4 + 4 4, where there is no operand between the final two 4s. Since this expression is incomplete, the player draws two cards (to bring them back to five in their hand) and play passes to the next person. The second player then plays two ÷ cards: the first ÷ over the + and the second ÷ between the last two 4s. This changes the expression to 4 × 4 ÷ 4 ÷ 4. Since this equals 1, the second player wins the round. Since there are four cards on the board (the two ÷ just played, the ×, and the + under one of these cards), the player also wins four points. The players discard or keep their cards, as they wish.
Round 2 begins with the next player, who plays two ÷ and a + card to form the expression 4 ÷ 4 + 4 ÷ 4, which is equal to 2. This player wins round 2 with 3 points. Round 3 begins with the next player, etc.
Optional Rules
Players may decide to include the following optional rules.
Implied Multiplication: Two expressions separated by parentheses (or a parentheses preceding or following a 4) implies multiplication. Thus 4(4 + 4) / 4 = 8 is allowable.
Subtraction as Negation: The subtraction card (-) can be used to indicate negation. Thus -4 + 4 + 4 + 4 is allowable. Multiple negations must be enclosed with parentheses: --4 is not allowed, but -(-4) is.
Extended Rounds: The game must end after the 9th round because it is impossible (so far as know) to form 10 using four 4s and the standard operations of arithmetic. However, you can do so if you allow a broad interpretation of the number 4. In particular, if you allow two adjacent 4s to represent 44, then you can find 10 = (44 - 4)/4. I haven't playtested this, and I suspect that this rule is unstable (if you place an operand between two adjacent 4s, it will take at least two more plays before it can be eliminated).
Appendix: The Four 4s
In case you're having difficulty making a particular number, here is a set of solutions.
1: 4 ÷ 4 × 4 ÷ 4
2: 4 ÷ 4 + 4 ÷ 4
3: (4 + 4 + 4) ÷ 4
4: 4 + (4 - 4) × 4
5: (4 × 4 + 4) ÷ 4
6: 4 + (4 + 4) ÷ 4
7: (4 + 4) - 4 ÷ 4
8: 4 + 4 + 4 - 4
9: (4 + 4) + 4 ÷ 4