Decimal of fortune What follows is a description of a game for two people, Playe

Question

Decimal of fortune What follows is a description of a game for two people, Player A and Player B determine anumber that has been selected The object of the game is for Player B to by Player A (and is unknown to Player B as the game begins). A score is computed based on how close Player B has come to Player A's number at the end of the game. Play requires a calculator Player A writes down a decimal with eight decimal places. This decimal must be between zero and one. Player B enters the value zero into the calculator. Player a number from List E (evens) or List o (odds). If B will begin play by selecting the number is chosen from List E, it is added to the value in the If the number is selected from List 0, it is subtracted from the value in the calculator. List Eis the infinite list of numbers: 1/2, 1/4, 1/6, 1/8 List o is the 1/2n, infinite list of num bers: 1,1/3, 1/5, Player A then tells Player B 1/(2n-1), whether or not the value in the calculator is greater than Player A's selected number, less than Player A's selected number, or equal to Player A's selected number. If the value in the calculator is equal to Player A's numal ber, play terminates. If not, Player B selects another number that has not been selected before, from either List Eor List 0 and adds it to the value in the calculator if it is from List E or subtracts t from the value in the if it is from List o in an effort to get closer to Player A's number. Play continues until Player B determines Player A's numal ber or until Player B has chosen 20 numbers from the lists. (Player B may not use any number from the lists more than once, but may choose numbers in any order.) the latter case, Player B may make a final guess (without further calculations on the calculator) Player B's score is 20 points if Player A's number is determined exactly; otherwise it is n points if the value guessed matches n decimal digits exactly. Notes: Player B needs to keep track of those numbers that have been used. It is suggested that numbers be written down as they are used while play is in progress. It is also suggested that values in the be written down (or stored memory) as they appear, so that if a mistake is made entering a number the previous may be recovered. For exam ple, a game where Player A chooses 0.42000000 (unknown to the next page. Player B) is given on Play this game four times, wit th each partner assuming the role of Player Atwice. Then answer the following questions. i. What strategies were developed for Player B as the games were played? 2. What strategies were developed by Player A to prevent Player B from deter- mining Player A's number? 3. the game were to continue "indefinitely," do you think that Plat number could be determined exactly? or why not? yer A's Copyright 1994 John Wiley & Sons, Inc. Mond

Explanation / Answer

Solution:

1. Strategies adopted in favour of Player B:

_An infinite list of both odd and even numbers has been provided. Therefore there is a wide range of numbers to choose from, in order to find out the number.

_An odd number when chosen, is subtracted from the previous determined number, whereas an even number is added with it. Therefore this strategy is helpful for narrowing down the range, i.e finding out the upper limit and lower limit within which the number must lie.

_The numbers from the list can be chosen in any order, therefore there is a chance to determine the narrowest range containing the upper limit and the lower limit quite quickly, if chosen wisely.

_A total of 20 numbers can be chosen, or there are 20 chances to find the number nearest to the number as selected by Player A. Hence the probability of finding out the exact number, or the number correct upto some places of rhe decimal is quite high.

_When 20 chances are over, Player B will be given a chance to guess the number without using the calculator. Depending on how narrow a range he/she had obtained, Player B has a chance to at least, tell the exact number selected by Player A.

2. Strategies adopted against Player B:

_The numbers in both the even and odd lists constitute of only inverses of even and odd numbers, respectively. In the odd list there was a scope to provide fractions with other numbers than 1, to function as numerator. This could have helped Player B find out the exact number more easily, quickly, and accurately.

_No number from either list can be used more than once. Therefore, once a narrow range has been obtained, there would be some difficulty to fine tune the selected number accurately to all places of the decimal, even if Player B has got an idea of the exact estimate of the number.

_Since no number from the lists can be repeated, 20 chances of finding out the selected number correct upto all the 8 decimal places is much difficult.

3. Even if the game was to continue indefinitely, there is a greater chance that the number as chosen by Player A cannot be chosen correct upto all the decimal places. This is because, since no fraction from either list can be used more than once, the scenario could be such that, the number which when added, or subtracted could have given the exact answer, has already been used up before. Although the numbers can be chosen in any order, the fact that odd numbers are subtracted, while even numbers are added, will make it difficult to reach the exact number correct upto all decimal places. Even if a close range of fractions are added or subtracted strategically, there will always be some deviation.

4. A successive combination of "too high, too low" can very effectively help to find out the upper bound or, how far off Player B is. In fact, it will actually help to fine-tune the range in the beginning, thereby reaching the number closest to the number as selected by Player A.

Related Questions

Navigate

• Browse (All)
• Browse D
• Subjects

---

---

Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.