What’s more fun than playing a game? Making your own game! One of our favorite preschool games is Memory, the matching card game. My son usually plays the official version at Grandma’s house, but we don’t have that game at home. The concept is so simple that we decided to make our own with readymade stickers and cardstock.
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The stickers that come with many stickers to a sheet, with repeating images are ideal, like the ones sold in the education/teacher aisle at your local dollar store. Or, buy two packages of identical stickers. Any kind of unmarked cardstock will work. The heavier, the better!
Project Materials:
- Stickers (you must have 2 of each image), $1
- 1-2 sheets of cardstock or heavy paper, $.50 or less
- Paper cutter or scissors, on hand
Total cost: $1.50
To Make:
You will need one card per sticker. Determine how many cards you will need based on the number of stickers you have.
Cut cards from cardstock. To cut 16 cards from one sheet of paper, cut paper in half from both directions. Cut resulting rectangles in half again in both directions. This is easiest if you have a paper cutter, but you can also do it with scissors.
Stick stickers to cards. Your child can help with this (my two year-old did a pretty good job! The images weren’t perfectly lined up on the card, but he didn’t care, and he had a ball
sticking the stickers on the cards).
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Note: it is best if stickers are different-looking enough to not be confusing to little ones. In other words, two different shots of the same breed of doggie might be a little too similar! We
used these stickers that all have a different baby animal on them.
Play the game!
To Play (ages 3 and up):
- Mix cards up.
- Lay all cards face-down on table (in a grid pattern).
- Youngest player goes first and turns two cards of her choosing
over. If cards match, she takes the cards and has another turn. If
cards don’t match, move to the next player. - When all cards are matched, the player with the most sets of matching cards wins.
To Play Simplified Version (ages 2 and under):
- Limit total number of cards to 8, or four sets (you adjust for your child’s skill and interest level).
- Youngest player goes first and turns two cards of his choosing
over. If cards match, she takes them, but does not have another turn. - Next player takes a turn.
- When a match is made, everyone cheers.
- When all cards are matched, start over again.
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The Not-Ready-For-Organized-Games version (younger 2s and under):
- Turn 4 sets of cards face up.
- Take turns looking at each card and trying to find its partner.
- Cheer when child finds a match.
Other tips:
- You can also use these cards to play other games like Go Fish.
- If you don’t have stickers that match, how about making your own? Print photos of your family (faces are good) on labels and stick onto cards. Avery has free label-making software that makes it super easy to print straight to labels! This would also be a fun handmade gift for a birthday boy or girl.
Heather Mann is a regular contributor at Make and Takes. She’s is the mother of two boys under age 3, and another boy on the way. She publishes Dollar Store Crafts, a daily blog devoted to hip crafting at dollar store prices, CROQ Zine, a print magazine devoted to hip crafting, and also CraftFail, a community blog that encourages crafters to share their not-so-successful craft attempts.
The card game memory, aka concentration, is a children’s game that tests visual recall ability.
There are n pairs of cards face down. The first player flips over 2 cards to make a pair. If a match is made, the player keeps the cards, gets 1 point, and goes again. If not, the player flips the cards over and the next player goes, playing by the same rules. The game ends when all pairs are made and the person with the most points wins. (In this version, the goal is to maximize the point total).
In 1993 the best strategy for the game was solved, assuming players have perfect memory (they remember every card that was flipped) and play a zero sum game to maximize the expected number of matches (+1 means match for you and -1 means match for opponent).
It turns out there are some surprising moves: sometimes you will want to sacrifice your turn by intentionally not making a match! The reason is it can be strategically smart not to flip over new cards in some situations to prevent your opponent from gaining information.
I made a video that explains the best strategy and whether you want to go first or second.
I summarize some of the points below.
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'All will be well if you use your mind for your decisions, and mind only your decisions.' Since 2007, I have devoted my life to sharing the joy of game theory and mathematics. MindYourDecisions now has over 1,000 free articles with no ads thanks to community support! Help out and get early access to posts with a pledge on Patreon.
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Types of moves
In a turn you flip over 2 cards. The moves are characterized by the maximum number of unknown cards you will flip.
The way most people play the game is by using “2-moves.” You flip over an unknown card and try to match it with one you have seen. If you cannot make a match, you then flip over a 2nd unknown card searching for a match.
Sometimes it is smarter to make a “1-move.” You flip over an unknown card and try to match it with one you have seen. If you cannot make a match, you flip over a known card that is not a match. In other words, a 1-move has a maximum of 1 unknown card turned up.
There is also the “0-move,” where you flip over 2 known cards that do not match! That is, you flip over 0 unknown cards. This amounts to passing on your turn. There are positions of the game where both players prefer a 0-move to avoid making a mistake and revealing a card location to the opposing player. The game essentially ends in a draw since no player wants to flip over an unknown card.
Example: 3 pairs of cards, 1 known
I show this in the video better, but let me try to explain it in text.
Suppose you have 3 pairs cards A, A, B, B, C, C shuffled on a table. You know the first card is an A. You flip over the second card as B. What is your best move?
Most of us would search the remaining 4 cards to find a match for B. This is a 2-move. What is the conditional expected value of completing a 2-move? There are 3 cases to consider.
–In 2/4 cases you get a C card and lose your turn. Now your opponent knows the location of A, B, and C, and each of the remaining cards will match. So your opponent will easily run the table and make 3 pairs. This is a disaster for you. Your expected value is -3.
–In 1/4 cases you get a B, so you get 1 point for sure. Then you are left with 4 cards, of which you know the location of one A. The value to the game will be X.
–In 1/4 cases you get the other A. You lose your turn, and you have guaranteed your opponent pairs the A cards (-1 to you). Then your opponent has 4 cards, of which he knows the location of one B. The value to the game will be -X since this is the mirror situation of the last case, but your opponent is playing.
The second and third cases have opposite expected values so they net to 0.
The expected value of a 2-move is therefore (2/4)(-3) = -1.5.
This is not good: you are expected to make 1.5 fewer pairs than your opponent.
What if you did a 1-move instead? You give up your turn by flipping over the A. Now your opponent has 6 cards and knows the location of one A and one B.
What’s the conditional expected value?
–In 2/4 cases your opponent matches either the A or the B, so it’s a -1 to you. Then the game has 4 cards, of which one is still known. I will skip the details, but there is a 2/3 chance your opponent makes both of these (-2), and then a 1/3 chance they mess up so you make these (+2).
–In 2/4 cases your opponent gets a C. If the opponent uses a 1-move, then you will run the table and get all 3. So instead the opponent will use a 2-move and hope to find the other C card. There is a conditional 1/3 chance of that, and then the opponent runs the table (-3). There is also a conditional 2/3 chance of messing up, in which case you run the table (+3).
The expected value is therefore:
(2/4)[-1 + (2/3)(-2) + (1/3)(+2)] + (2/4)[(1/3)(-3) + (2/3)(3)] = -1/3 = -0.333…
A 1-move is actually better! Your loss is -0.333 versus that of a 2-move which is -1.5.
Optimal strategy
The game has been solved by computer (Zwick and Paterson 1993). The optimal strategy is defined in terms of the pairs of cards n and the number of known cards k. Here is how to decide which move to make.
Play a 2-move if k = 0 (there is no other choice) or if n + k is odd. There is one special case exception: for n = 6 and k = 1 you should do a 1-move.
Play a 1-move if k ≥ 1 and n + k is even. Also do a 1-move for n = 6 and k = 1.
Play a 0-move if n + k is odd and k ≥ 2(n + 1)/3. This happens when the number of known cards is relatively large to the number of pairs. The intuition is you don’t want to risk flipping a card since you might make a mistake and your opponent will then make many pairs. An example is if you have 5 pairs and there are 4 known cards.
Playing first or second?
The paper also analyzed the expected value to the game.
If n is 1, 6, or an odd number 7 or larger, it is best to go first. The expected value to the first player is about 1/(4n + 2).
Otherwise, it is best to go second. If n is 2, 3, 4, 5, or an even number 8 or larger, it is best to go first. The expected value to second player is about 1/(4n – 2).
This is a zero sum game, so the expected value to the other player is the negative value.
Note the expected value is roughly 1/(4n). A fair game has a value of 0 to both players, so the game of memory is almost fair with a slight edge to one of the players.
Now you know the best way to play and whether you should go first or second, so go out and dominate some people in the game of memory!
Credits
U. Zwick and M. S. Paterson, The memory game. Theoretical Computer Science, 110 (1993):169-196
http://www.sciencedirect.com/science/article/pii/030439759390355W
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SPECIAL THANKS TO Possibly Wrong! Check out this excellent post that provides details and references to other papers on this topic as well: https://possiblywrong.wordpress.com/2011/10/25/analysis-of-the-memory-game/
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I read about the 3 pairs, 1 known example from the chapter on Memory in the book “Luck, Logic, and White Lies: The Mathematics of Games” by Jorg Bewersdorff. http://amzn.to/VIHHkt
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