Krypto: A Computation Game

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Krypto is a commercially produced game consisting of cards with numbers on them. Five cards are dealt face down to each player. A single card is dealt face up, and this one card becomes the object card for each player. These are the rules:

  1. 1. Each player must then attempt to use the number on each card in her or his hand to form an expression that has the same value as the number on the object card.

    2. Each card in each hand must be used once and only once in the expression.

    3. Any combination of addition, subtraction, multiplication, and division may be used in forming the expression. (Note: This does not rule out the possibility of using the same operation throughout an expression.)

    4. The first one in a group to get a correct solution wins.

Example:

  1. Hand: 5, 9, 11, 2, 24

    Object: 21

    Solution: (5 + 2) x [(24 + 9)/11] = 21

 

The purposes are the same as in the game of Nim:

  1. 1. To build interest in mathematical thinking
  2. 2. To develop skills in mathematical thinking
  3. 3. To provide practice in developing and using cognitive strategies
  4. 4. To provide practice in addition, subtraction, multiplication, and division in a context other than pure drill

I. Solving the Problem

Each hand is a problem to be solved. All students need to learn to ask themselves the questions that will lead to the solution. A good way to start is to simply begin trying the numbers in various combinations, and often a solution results. Solving a hand is not like, say, doing a multiplication example for which you already know all the steps to take, and in which order to take the steps.

II. A Cognitive Strategy

A useful cognitive strategy for solving the Nim games is "Working Backwards". A helpful cognitive strategy for solving a Krypto hand is "Divide and Conquer". That is, separate the problem into parts, solve each part, and then fit the parts into the total solution.

Example 1:

Hand: 24, 12, 16, 8, 4

Object: 5

Divide the numbers into two groups: 24, 16, 8 and 12, 4

If you could compute a 2 from the green group and a 3 from the gold group, you could add the 2 to the 3 to get the object 5.

If one separation does not work, you try a different one. You may never find a solution, but think of all the practice in arithmetic you get while you are trying.

Example 2:

A better grouping might be to place 4 in one group and 1 in the other group.

Hand: 11, 22, 9, 8, 3

Object: 12

Divide the numbers into two groups: 8, 22, 3, 11 and 9

If you could get 3 with the green group, you could add it to the 9 to get the object 12.

If you could get 21 with the green group, you could subtract the 9 to get the object 12.

If you could get 108 with the green group, you could divide it by 9 to get the object 12.

So, the problem is simplified in two ways:

  1. 1. You have fewer numbers to work with at any one time.
  2. 2. You have more object cards to work toward.

This increases your chances of finding a solution if one is possible.

In this case, you can get a 3 from the green numbers like this: (22+11)/(8+3).

Also, you can get a 21 like this: 22-(8+3)/11).

I don't think you can get 108 from the green numbers, but you never know until you try all possible combinations. This is what makes the game such a good motivator of computational practice.

III. Can all possible hands be solved?

The answer depends on what numbers are in the deck of cards. For example, consider the following possible hand:

Hand: 1, 1, 1, 2, 2

Object: 25

Using only some combination of addition, subtraction, multiplication, and division, this hand could not be solved.

Question: What is the largest number you could get with this hand?

IV. At what level can this game be used?

This game can be used at any level from fourth grade on up. All that is required is a working knowledge of addition, subtraction, multiplication, and division. There is no upper limit to the use of the game. Some of the hands can be challenging to adults, even to mathematics majors.

V. How can this game be adapted to children in lower grades?

  1. 1. Fewer cards could be dealt in each hand.
  2. 2. Only addition and subtraction could be allowed.
  3. 3. Only small numbers could be used.

VI. What are some variations of the game?

1. Use dates expressed in digits. One rule might be to use the digits in the order they appear with any combination of addition, subtraction, multiplication, and division, and put the equals sign anywhere you wish.

Example 1: September 17, 1997 (9/17/97)

Example 2: September 19, 1997 (9/19/97)

2. Use four 4s to make an expression equal to as many whole numbers as you can. Be creative in making your own rules for this game.

Example 1: (4 + 4) - (4 + 4) = 0

Example 2: (4 + 4) / (4 + 4) = 1

Example 3: (4 x 4 x 4) / 4) = 16

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