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Name: Kevin Cheung
Age: 17
Home Town: New York City, NY

I originally wrote this paper while I was a senior in Stuyvesant High School. My intention was to submit this paper to the Intel Science Talent Search. The paper you will read is the exact work I sent to the committee. However, I was left behind, and my paper was deemed by the committee to be insufficient for me to become a semifinalist. Nonetheless, I still submitted my paper to a couple of local competitions. My project was chosem as a finalist in the Junior Sciences and Humanities Symposium, but didn't end up going any further.

I am the oldest of three kids, all male. It's still quite a wonder that I wrote this paper, since I am the only person out of the five in my home that watches it. Everyone else seems to detest it. I am now a freshman at NYU, and am on track to become a pre-med mathematics major.

Note 1: This project is far from done. If I so choose to pursue a higher degree in mathematics, I may just pick up this project from where I left off. If you have any comments, feel free to contact me at kevinkc@hotmail.com.

Note 2: I forgot to add one person to my acknowledgements. I'd like to thank Mr. Jim P. Balcerek for creating the mods I used in my research.
  1. Introduction

    1. Purpose

      This study is an analysis of the game "Press Your Luck" under the modified condition of the "perfect strategy." Because of some inherent factors in the game that do not guarantee a win or loss on a given move, the condition must be redefined so that "perfect strategy" must be the strategy that allows for the greatest probability that a given player will win. The analysis will identify in what situations each of the three players can win.

    2. Rules of "Press Your Luck"

      1. The game is played with three players.

      2. The game board consists of eighteen spaces, arranged in a rectangular 6x5 grid with the middle squares removed. Each of the squares has three different states, for a total of fifty-four different possibilities for a move. Surrounding each square is a ring of lights. Only one of these rings is active at anytime.

      3. About twice every second, another space is highlighted by the ring, so the cursor "bounces" around the board. The bounces are governed by a random-number generator. While the cursor is bouncing, the squares themselves change from state to state.

      4. Each player begins with a certain amount of spins, based on trivia questions they have answered beforehand.

      5. The game consists of two rounds. During the first round, the player with the least amount of spins begins. During the second round, the player with the least score begins. If there is a tie, the player to the farthest left begins.

      6. During each player's turn, they may choose to press their luck, which means to take a spin, or pass their spins.

      7. A spin is complete when a player hits the button on their podium. The board stops, and the cursor stops at a square, indicating what the player has won. This consumes one spin. The value of the spin is added to the player's cumulative score.

      8. If a player hits a whammy, that player's score is reset to zero. That player receives a whammy counter. When a player earns four whammies, that player is immediately eliminated from the game. Any spins that the player had are removed.

      9. If the active player decides to pass, all of the spins that the player has (which exist in an "earned column") go to another player's passed column, depending on the chart below. If there is a tie, the passing player may choose who receives the spins.

        • 1st place passes to 2nd place
        • 2nd place passes to 1st place
        • 3rd place passes to 1st place

      10. All of the spins in a player's passed column must be taken before that player can pass his spins.

      11. If a player hits a whammy when there are still spins in his passed column, all of the remaining spins in the passed column are moved to the earned column.

      12. When a player has exhausted all of their spins, that player's turn is over and the player in second place takes his turn.

      13. When all players have exhausted their spins, if it is the first round, the player in the lead goes last in the second round, followed by the player in second place, and finally the player in third.

      14. If it is the second round, the player who has the greatest total when all players have exhausted their spins wins the game.

    3. Squares

      The board contains several different types of squares, which are listed below.

      1. Cash- This adds the amount of cash shown to the player's total.

      2. Prize- The value of the prize is unknown until it is hit. The cash value of the prize is added to the player's total. On the board, the prize is replaced with a new prize.

      3. Cash + One Spin: This adds the amount of cash shown, and it adds an additional spin to the player's earned column.

      4. Move One Space: This allows the player to choose between the values in the squares that the arrows in the Move One Space square point to.

      5. Pick a Corner: This allows the player to choose from the values in the corner squares of the board.

      6. Advance/Go Back Two Spaces: This advances the cursor two spaces in the direction that the arrow is pointing. The player receives whatever is in the square that the cursor lands on.

      7. Big Bucks: This moves the cursor to the square where the highest amount of cash is held. This square is on the top row, fourth square from the left.

      8. $2000 or Lose One Whammy: This allows the player to choose a flat $2000, or to lose one whammy counter. This is often used when a player has three whammies, when they are at high risk of elimination.

      9. Whammy: Described in the previous section.

    4. The Board

      Although the boards changed from time to time, this board is taken from the fall 1984 episodes of Press Your Luck. If the board is numbered clockwise, from the top left corner, the configuration of board from the time is shown below. (Editor's note: you can view this board on this page. --CY)

      Round 1:
      1. $350, Move One Space (to square 2 or 18), Whammy
      2. $750, $500, Prize
      3. $400, $250, Whammy
      4. $1000, $1250, $1500
      5. $300, Prize, Whammy
      6. $525, $650, Go Back Two Spaces (to square 4)
      7. $470, Prize, Whammy
      8. $550, $450, $300
      9. $350, $550, Whammy
      10. $300, Prize, $500+One Spin
      11. $200, $600, Advance Two Spaces (to square 13)
      12. $400, Big Bucks, Whammy
      13. $500, $750, Prize
      14. $400, $500, Whammy
      15. $550, $700, Prize
      16. $300, $500, Whammy
      17. $200, Prize, Whammy
      18. $100+One Spin, $200+One Spin, $300+One Spin

      Round 2:
      1. $1750, $2250, Whammy
      2. $500, $1250, Prize
      3. $500, $2000, Whammy
      4. $3000+One Spin, $4000+One Spin, $5000+One Spin
      5. $750, Prize, Whammy
      6. $700+One Spin, Go Back Two Spaces (to square 4), Pick a Corner (to square 1, 10, or 15)
      7. $750, Prize, Whammy
      8. $500+One Spin, $750+One Spin, $1000+One Spin
      9. $800, Move One Space (to square 8 or 10), Whammy
      10. Prize, Prize, Prize
      11. $1500, Advance Two Spaces (to square 13), Whammy
      12. $500, Big Bucks, Whammy
      13. $2000, $2500, Prize
      14. $2000, Move One Space (to square 13 or 15), Whammy
      15. $1000+One Spin, $1500+One Spin, Prize
      16. $600, $750+One Spin, $2000 or Lose One Whammy
      16. Prize, $700+One Spin, Whammy
      17. $1400, $750+One Spin, $1000+One Spin

  2. Definitions

    1. Notation

      Throughout the study, the following diagram may be used to illustrate a position in the game:

      Player 1: 3 6 $1500 3       Player 2: 0 0 $0 4       Player 3: 2 0 $15429 0

      For each player, the numbers, from left to right, represent: earned spins, passed spins, score, and whammies. The player in bold is currently active and spinning.

    2. Expected Value

      The expected value for any given spin is the average amount of money a player will make on any given spin. To calculate expected value, a player takes the average of all of the prizes, cash, and directional spaces on the board and multiplies it by the probability of hitting such a space and subtracts the amount of money they would lose if the next spin was a whammy multiplied by the probability of hitting a whammy. In each round, the probability of hitting a whammy on any given spin is 1/6. So, the expected value of a spin (E) in each round is as follows:

      Round 1: 5/6(avg1)-1/6(S)
      Round 2: 5/6(avg2)-1/6(S),

      where avg1 and avg2 represent the average values of the non-whammy spaces in rounds one and two respectively, and S is the player's current score. The values of avg1 and avg2 vary with the prizes on the board, but their values are very close to $600 and $2000 respectively.

    3. Median of Values

      Although the value of E provides a good approximation of what the next spin will herald, the distribution of the values of the squares is not symmetric. Therefore, it is also useful to know the median value of the squares on the board. Assuming that the values of the prizes have a relatively symmetric distribution, the median is approximately $450 for the first round and $1400 for the final round.

  3. Final Spins

    Since "Press Your Luck" can be won or lost on the final spin, it is important to define strategies for those spins first.

    1. Position A1

      This position occurs when the active player is in third place and he has one earned spin left. One example is shown below:

      Player 1: 0 0 $15200 0       Player 2: 0 0 $5000 0       Player 3: 1 0 $500 0

      Player 3 must obviously press his luck. His only alternative would be to pass his spin to Player 1. If Player 1 does not hit a whammy, he has won; otherwise, Player 2 has won since he would be in first place at the end of the game.

    2. Position A2,1

      This position occurs when the active player is in second place, has one earned spin left, and is behind by less than the median value of the squares on the board. Here he should still press his luck, since the chances of making the deficit is better than the chances of failing if he takes the spin. If the active player passes, the probability of a win is only 1/6, meaning that the player in first place must hit a whammy.

    3. Position A2,2

      This position is exactly like position A2,1, but the active player is behind by more than the median. Similar reasoning can be used for this case. The active player should press his luck, as long as he is behind by less than E. If he is behind by more, it is in his best interest to pass his remaining spin. The worst-case scenario states that there are no prizes or cash amounts that can cover the deficit, and a spin must be earned if the game is to continue.

    4. Position A3,1

      This position is identical to A2,1 except that the active player is in first place and is leading by less than the median value. The active player should press his luck, since the gap is small and the probability of a win is close to 5/6, since any square besides a whammy could give him a win. If he were to pass, the chances of overtaking the first player is certainly greater than 1/6, since there are at least seven squares that yield more than the median.

    5. Position A3,2

      This position is identical to A3,1 except that the active player is ahead by more than the median. Here, passing is a good policy, especially if the active player is ahead by more than about $7000, since that is the upper bound for the value of prizes in round two. In the most extreme case, the probability of player two winning on one passed spin is zero, so passing is recommended. As the difference gets lower, the probability of winning starts to climb, but not any higher that about 3/14 assuming that all of the prizes have values over the median. In any of the positions above, if the spinning player hits money and a spin, the situation is redefined into one of the five already analyzed cases, and can be played from there.

  4. Directional Spaces

    Some of the most interesting features of the Big Board in round two are the use of directional squares, namely, the "Move One Space" and the "Pick a Corner". This usually provides the players with a strategic choice between money and a spin and either cash or a prize. Normally, if a spin is offered, it should be taken, since spins are the only way to earn any money in the game and have a chance at winning. A player with many spins also wields great strategic power in the game and has the capability to win a lot, or force a player to lose it all. There are some exceptions to this rule, especially if it is the final spin of the game. Since victory is valued over any spin, a prize or money should be taken if it assures victory. For example,

    Player 1: 0 0 $12200 0       Player 2: 0 0 $5000 0       Player 3: 1 0 $12561 0

    Assume that on the final spin, player three lands on "Move One Space" to $500+One Spin or a trip to Tahiti. Obviously, Tahiti should be taken, since taking the spin could cost you the game later on if a whammy appears. If a choice of a prize or money is given, opt for whatever seems like it is valued more. Usually, one can judge the approximate value of a prize by its name. For example, if a choice of $2500 or a car is given, the car is obviously worth than $2500, so it should be taken. As spins begin to dwindle, on the other hand, a spin may not be opted for if the active player has a favorable position. For example, if the active player is in first place and has a wide margin, the player may not choose a spin as to make the situation harder to come out of if he decides to pass.

  5. Passing Spins

    The true strategy of "Press Your Luck" lies in the passing of spins. These spins, by definition, must be taken, and can spell doom or success for the recipient. Therefore, it is imperative to know when to pass spins.

    1. Using Expected Value

      For the most part, a spin will get the active player money. However, as the scores rise towards the five-digit mark, players will begin to become hesitant to press their luck since one whammy can erase many spins of work. One of the simplest ways to know when to pass is to know the value of E. If E<0, obviously, the spins should be passed since taking one means risking too much money. It is simple to calculate when E<0.

      E=5/6(avg1)-1/6(S)<0
      -1/6S<-5/6(avg1)
      Dividing by -1/6,
      S>5(avg1). Similar reasoning can be used for the second round. Substituting the values for avg1and avg2 yield that if a player has over $2500 in round one and $10000 in round two, he should strongly consider passing. Of course, this may not be the best course of action in certain situations.

    2. "Lockout"

      Players in first or second place may invoke this theory in planning strategy. Let us assume that this situation occurs in round two:

      Player 1: 3 0 $7200 0       Player 2: 0 0 $5000 0       Player 3: 1 0 $6514 0

      Since it is player three's turn, he has the option to press or pass. If he presses, and hits a whammy, the game is all over for him. He will drop to third place, and will never have a chance to recover. Since he would be in third place after the whammy and have no spins left, he would not have a chance to earn money on his own. Spins will never be passed to him since he is in third place. Since passed spins can only be received by the players in first and second place, player three simply becomes a spectator to the game. This is the essence of the lockout argument.

      Proof:
      Assume that player 1 is in second place and has some amount of spins left in his earned column. The hypothesis states that should he pass, the spins will never reach a certain player, say player 3, in third place. By rule #10, player 1's spins will be passed to player two, in first place. If player two does not whammy before he decides to pass any of his spins back, he will obviously be in first place and by rule 10 will be passed to player 1. If player 2 should whammy, he must press his luck to remain in the game. If he passes, the other players will be the only ones with a chance to win, so player 2 will press his luck until he exhausts his spins, or ends in second place and passes back to player one. A similar argument can be used if player 1 whammies. Thus, player 3 will never receive spins for the duration of the round, and will lose the round.

      So using the lockout argument, player three should pass, so spins will never get to player two and therefore locks player two out of the game. This argument can also be used for players in first place. For example:

      Player 1: 0 0 $1600 0       Player 2: 0 0 $5000 0       Player 3: 4 0 $12561 0

      Playing lockout can also be used as a defensive strategy, especially in the first round. The tactic is most useful when a player accumulates two whammies and still has spins left to play. In round one, the scores only determine turn order, so having a high score is not as important. After hitting the second whammy, passing is very much advised, since hitting a third whammy in round one severely hampers one's chances for making much money in the final round. Even worse, if a player has three whammies, that player is at high risk for elimination in the first round. Although the chances of such an event occurring are rather low, the risk of being crippled for the rest of the game is sufficient to merit passing. Going first in the final round is obviously better than being eliminated from the game prematurely.

    3. Whammies

      Having three whammies is a serious burden on a player. One spin could mean elimination. Therefore, a player having three whammies will want to pass as soon as possible. The converse is also true. A player in first place will want to pass spins to this player, hoping to capitalize on their weak position. However, the player in this predicament will want to consider the other theories and make sure that after passing, they will still have a chance to win. For example, the whammy count for a player is irrelevant if it is the final spin of the game. If that occurs, then the players can proceed as was described earlier. In addition, a player with three whammies in third place in the second round won't pass, since passing is just as bad as whamming out of the game. Obviously, landing on $2000 or Lose One Whammy is a life-saver, taking that player out of the danger zone. In the first round, assume that a player has two whammies and still has spins left in the earned column. As described in the previous section, these whammies are a serious burden, since hitting a third whammy in round one severely hampers that player's progress in the second round. Therefore, passing is extremely recommended since the threat of the third whammy in round one is simply too great.

    4. Multiple Spins

      As stated before, spins are the only way to gain money in the game. Naturally, the more spins a player has, the more money they are liable to make. In addition, the existence of many passed spins in a player's passed column is almost disastrous to that player. Since the probability of hitting a whammy on a given spin is 1/6, naturally, out of six spins, it is likely at least one of them is a whammy. Therefore, if a player has six or more spins and has a commanding lead, he should consider passing, since it is likely that the recipient will hit a whammy. This is especially useful when the recipient has three whammies, since hitting a whammy means that the passer doesn't have to worry about getting spins passed back to him from the recipient.

    5. The Safety Net

      "Press Your Luck" is well known for its randomness and its tendency to be unpredictable. Since it is extremely hard to analyze positions even three spins ahead because there are so many possibilities, it is often necessary to have some sort of safety when playing through the rounds. One of the most basic principles of a safety is the creation of a "safety net." For "Press Your Luck," this consists of a supply of spins to use if a player hits a whammy. For example, a player with a plethora of spins and either zero or one whammy can continue to press his luck beyond the levels prescribed before, since he can easily make it up with the remaining spins. However, as a player's score increases, the amount of money to be replaced increases and the supply of spins dwindles. Therefore, it is important for players to gauge the amount of money that is required for each spin on average if the player does hit a whammy. The expected value is a good indicator of the money that can be made per spin. The active player should often consider this value s/(e-1), where S is the player's score and e is the number of spins that the player has. If this ratio is higher than E, it is likely that, assuming a whammy on the next spin, the player will not make up the difference, and passing should be brought into consideration. If the ratio is less than E, the player can consider continuing to play, knowing that the money can be relatively easily recouped in the following spins.

Possible Directions of Future Research:
In this study, we give guidelines for a strategy when playing "Press Your Luck." Future research could include programming this strategy and running computer simulations of the game to test its effectiveness.

Acknowledgements:
I would like to thank Mr. Bruce Winokur, my Intel Mathematics teacher, for giving me guidance and giving comments for the first drafts of my paper. In addition, I would like to thank Mr. Curt King and all those at Doulos Software, who programmed an excellent computer simulation of Press Your Luck which I used to test and refine my strategies. Note: The application can be downloaded for free at http://www.crossbearer.com.

This paper is the property of Kevin Cheung, copyright 2003. This document may be used for private purposes only. It may not be distributed publicly, in whole or in part, without the explicit written permission of the author.

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