Mathematical games are always a popular adjunct in any teacher's repertoire of motivational activities. This game promotes learning and fun at the same time by combining two unlikely allies: a deck of ordinary playing cards and a calculator. The material presented here utilizes a simple, four-function calculator with a square root key. With minor adjustments, the more powerful scientific models may be used.

     There are many ideas that a student can pick up while playing this game, not the least of which is the added familiarity with the calculator's utility as a tool. Reading and writing the displayed digits carefully and precisely are skills that need to be developed in certain students. Of particular relevance to the game is the careful identification of which digits produce one's best hand.

     The basic game involves 3 players using 3-digit numbers. Remove the king, queen, jack and ten of any particular suit from the deck of cards. The nine remaining cards (Ace to 9) constitute the playing deck. The value of the Ace will be 1. Each player uses his/her own calcuator, a recording sheet with headings: Number, Root, Combination, Points, a copy of the scoring chart (Figure 1), and a pencil.


  1. The dealer shuffles the deck and deals 3 cards face down to each player.

  2. Each player arranges his/her cards left-to-right, without turning them over, in any way desired.

  3. When all players have arranged their cards, the dealer says, "Turn over." All cards are then turned face up without changing their left-to-right order. The 3-digit number thus produced is entered into the first column of the recording sheet.

  4. All players enter their numbers into their calculators and press the square root key. The displayed value, which represents one's playing hand, is written in the next column of the recording sheet.

  5. Each player then carefully studies the digits of the square root to determine the highest scoring combination (see Score Chart) which is then written in the third column. For example, the number 142 yields the square root of 11.916375, and the three ones give the highest scoring combination of "three of a kind".

  6. The points are determined by adding the base score given in the Score Chart to the value of the largest digit involved in the combination. The largest digit in the combination constitutes the Bonus Points. So, in the illustration above, 142 has a square root of 11.916375, for which the base score is 30 points (three of a kind: 1, 1, 1) plus 1 Bonus Point for the largest digit in the combination. After adding, the total (i.e. 31) would be written in the Points column of the recording sheet.

    Examples: If one has two pairs (6,6 & 2,2), the score would be 20 + 6 = 26 points.

    If one has a full house (4,4,4 & 7,7), the score is 50 + 7 = 57 points.

    For a run (1,2,3,4,5), the player scores 100 + 5 = 105 points.

  7. For the 2nd and 3rd hands of a game, Rules 1-6 are followed as before. But for the 2nd hand, the square root key is pressed twice; for the 3rd hand, the square root key is pressed three times.

  8. After 3 hands are completed, the points are totaled and the winner is declared.

A Sample Game
Number Root Combination Points
795 28.195744 1 pair: 4, 4 14
641 5.0316972 all different 200
436 2.1376461 2 pairs: 6,6,1,1 26

     This player's score is 240, a good game.


     First, from time to time a perfect square is formed by the cards. Since here a square root (i.e. one press of the key) only has 2 digits and that isn't enough to form any hand in the first part of the score chart, an award of 100 points is given. And once in a while, all the eight digits that appear in the display are different, or distinct. We feel that such an event is unique enough to merit the award of 200 points! Experience has revealed that occasionally only 7 digits (or even 6) appear, all of which are distinct. So a point value is assigned to each of these cases. However, no bonus points are given in any of the above cases.

     When using scientific calculators with 10-digit display capacity, just instruct the players to copy the first 8 digits that appear, that is, "truncate to 8 digits". While rounding the decimal part to achieve 8 digits is possible, our experience has shown that truncation is simpler to explain to the majority of students.


     There are many possibilities to vary the basic game: use more number cards at once (i.e. 2 sets of Ace through 9), deal more cards for each hand (i.e. 4 cards to form 4-place numbers), include one or more jokers (they equal zero, or become a "wild card"), allow more hands to constitute a game, etc. Presented below are several more variations that have proved popular and instructive in our classroom. Of course, the reader is encouraged to devise his/her own rules.


     An important spinoff can involve a statistical analysis of the data that can be collected after many games have been played by the whole class or merely a small group. For example, have the students compute the arithmetic mean of various final scores: per hand, per game, per group of players, etc. An additional connection to probability can also be explored by constructing frequency histograms of the types of the hands that occurred throughout. The raw data is easily obtained from the students' recording sheets.

     One of our students showed us an important example of that "what-if" curiosity that needs to be promoted more in our classes. She thought: "What if I had arranged my cards differently? What might have been my score?" So she took the card numbers from several of her games that day and proceeded to form all the various possibilities, then obtained the square roots and points. She was so proud of herself, as we were of her. What makes this more outstanding is that she was a below average performer, yet did this quite independently.

     This game, in its own special way, provides a connection between the world of regular mathematics and that of simple gaming pastimes. It's fun and one learns all the while.


           Combination                          Base Score

	One Pair                                  10 points

	Two Pairs                                 20 points

	Three of a Kind                           30 points

	Three Pairs                               40 points

	Full House (3 of one kind, 2 of another)  50 points

	Two Trios (2 sets of 3 of one kind)       60 points

	Four of a Kind                            70 points

	Five of a Kind                            80 points

	Run (5 consecutive digits)               100 points

	Special Cases:

	*Perfect Square                          100 points

	*All 8 digits different                  200 points

	*Only 7 digits showing, all different    150 points

	*Only 6 digits showing, all different    125 points

	[*No bonus points in these cases.]

				Figure 1

Sample Hands & Scores
Number Root Hand Points
158 12.569805 5, 5 15
145 12.041594 4, 4 & 1, 1 24
142 11.916375 1, 1, 1 31
138 11.74734 1, 1, 4, 4 & 7, 7 47
149 12.206555 5, 5, 5 & 2, 2 55
152 12.328828 2, 2, 2 & 8, 8, 8 68
485 22.022715 2, 2, 2, 2 72
147 12.124355 1, 2, 3, 4, 5 105
169 13 square 100
786 28.035691 all different 200

[Note: The square roots in this table were obtained from a simple, 4-function, 8-digit calculator.]

This article of mine is from MATHEMATICS TEACHING in the MIDDLE SCHOOL. NCTM. Feb. 1998.
pp. 366-8. Reprinted with permission. (See photos following.)

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