Yusuf Sila: Logic

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LOGIC EXERCISE

8:46 PM

Muhammad Yusuf , Posted in Logic , 0 Comments

1. If there are 2 apples, 1 orange, and 3 peaches in the basket and the same rules from the example apply, which type fruit will be left in the basket?

2. In the Stanley Cup playoffs, two teams play a maximum of 7 games against each other and the team that wins 4 games moves on to the next round. After a team has won its 4th game, no more games are played, for example, if the Leafs win 2 games, the Senators win 1 game, and the Leafs win 2 more games for a total of 5 games, then the Leafs will move on to the next round and the 6th and 7th games will not be played.

(a) If the Leafs won the first two games against the Senators and the Sens won the third game, construct a game tree to find all of the possible outcomes of the rest of the round.

(b) If each of the outcomes listed in part (a) are equally likely to occur, what is the probability that the Senators will win the round? (Hint: the probability that an event will occur is the number of ways that the event can occur divided by the total number of outcomes.)

(c) What is the probability that the Leafs will win the round in six games or less?

3. Andrew and Bethany are playing a game where they take turns taking marbles out of three jars. Andrew goes first. For each turn, the player must take the same number of marbles out of any two jars (for example, if Bethany takes two marbles out of one jar, she must also take two marbles out of another jar). The person who cannot make any more moves loses (ie. the person who is left with two or more empty jars loses). The first jar has 1 marble in it, the second jar has 2 marbles and the third jar has 5 marbles. Draw a game tree to find all of the possible outcomes. If each of the outcomes you found has the same chance of occurring, then who is more likely to win?

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Making Sense of Exponential Growth

1:29 AM

Money on Chess Squares

A chessboard has 64 squares. If we are going to place 1 cent in the first square, 2 cents in the second square, 4 cents in the third square, and so on, how much money do we need to fill all the 64 squares?

From the pattern above, it is clear that the amount of money placed in each square is twice than that of the amount placed in the preceding square. If we are going to number the squares from 1 through 64, then the amount of money needed to be placed in each square is shown in the table below.

As we can see from the table, the amount of money in the 64^th square is184,464,625,987,328,000.00. If we have indeed placed the money, the total money on our chess board is

Surprised?

How large is our money?

In US Dollars

The current world population, as of this writing, according to the US Census Bureau is approximately 6.8 billion. Dividing the total amount of money on our chessboard by the total world population, we can see that each person will receive $27, 127, 564.81. That makes everyone a millionaire.

According to Wikipedia, the federal budge for 2010 of the US is amounting to 3.55 trillion dollars. If we are going to use all the money on our chessboard, the US will have enough budget for 51,962 years.

In Philippine Pesos

Assuming that the money is in Philippine pesos. In 2009, the government budget of the Philippines is approximately 1,300,000,000,000.00 (1.3 trillion pesos) for the entire year. That means that if we actually have that amount of money, and suppose that we maintain the yearly budget to 1.3 trillion, we will have more than enough budget for the next 141,898 years!

How about rice grains?

If instead of 1 cent, we place 1 grain of rice in the first square, 2 grains in the second square, 4 grains in the third square, and so on, the total number of grains placed on

the entire chessboard is 2⁶⁴ − 1 = 18,446,744,073,709,551,615 grains of rice (see Geometric Series) or approximately weighs 461,168,602,000 metric tons (461,168,602,000,000 kilograms), the heap of which is larger than Mount Everest (Source: Wikipedia).

Now, the average life span of a Filipino is 70 years (25, 550 days). If each person eats half kilo a day (that’s an overestimation, I think it’s only a third of a kilogram), then the average rice consumption of a Filipino in 70 years is 12, 775 kilograms.

If we are going to divide 461,168,602,000,000 kilograms of rice by the average lifetime consumption of a Filipino, the answer is 36,099,303,483. That means that the total amount of rice can feed more than 36 billion people in their lifetime. As we have mentioned, the population of the world as of this writing is only 6.8 billion. If all the people in the world have the same average life span and the same average rice consumption as Filipinos, the rice on our chessboard will feed the world for approximately 370 years!

The examples above show the unimaginable rapid growth in numbers. Both examples shows the numbers 2ⁿ where n are integers. This fast growth is known as exponential growth.

Note: I have noticed in the 51^st number that the decimal place became 0.20 instead of 0.24. I think this is a MS Excel error. It’s probably having a hard time handing large numbers. The error though is not that significant in our analysis and computations.

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GAME THEORY PROBLEMS

12:24 AM

Muhammad Yusuf , Posted in Logic , 0 Comments

If there are 2 apples, 1 orange, and 3 peaches in the basket and the same rules from the example apply, which type fruit will be left in the basket?
In the Stanley Cup playoffs, two teams play a maximum of 7 games against each other and the team that wins 4 games moves on to the next round. After a team has won its 4th game, no more games are played; for example, if the Leafs win 2 games, the Senators win 1 game, and the Leafs win 2 more games for a total of 5 games, then the Leafs will move on to the next round and the 6th and 7th games will not be played.

If the Leafs won the first two games against the Senators and the Sens won the third game, construct a game tree to find all of the possible outcomes of the rest of the round.
If each of the outcomes listed in part (a) are equally likely to occur, what is the probability that the Senators will win the round? (Hint: the probability that an event will occur is the number of ways that the event can occur divided by the total number of outcomes.)
What is the probability that the Leafs will win the round in six games or less?

Andrew and Bethany are playing a game where they take turns taking marbles out of three jars. Andrew goes first. For each turn, the player must take the same number of marbles out of any two jars (for example, if Bethany takes two marbles out of one jar, she must also take two marbles out of another jar). The person who cannot make any more moves loses (ie. the person who is left with two or more empty jars loses). The first jar has 1 marble in it, the second jar has 2 marbles and the third jar has 5 marbles. Draw a game tree to find all of the possible outcomes. If each of the outcomes you found has the same chance of occurring, then who is more likely to win?

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MATH PROBLEM - LOGIC

5:15 PM

Muhammad Yusuf , Posted in Logic , 0 Comments

Fill in the boxes with the numbers 1 to 9 (the number 5 has been filled in for you). The value of each expression must equal the number at the end of the row or column. The operations are done in the order that they appear (ie. Ignore order of operations).