Professor Simonson History of Math Ingenuity

Fall 1997 MA 149

This is due in one week. The exam should be done individually and NOT in groups. You may come for help if you are stuck.

0. Preliminary Problems (20 points)

a. Given two jugs which contain 17 and 13 quarts respectively. How can you get 1 quart?

b. If x, y and z are the sides of a right triangle, prove that 5x, 5y and 5z are sides of a right triangle.

c. A friend of mine who went on a cruise mentioned that as the boat approached Boston, he could see the Prudential Center building 50 miles away. Consider the picture below and explain whether you believe him or not, and why. (The earth has a radius of about 4,000 miles; the Prudential building is about 600 feet high).

Earth

d. In the general game Nim, would you rather go first or second in a position where the are 20 rows each with 7 sticks in it? In general, if the number of sticks in each row is the same, then describe a simple way to determine whether or not you wish to go first.

1. Hebrew Number System (10 points)

The Hebrew Number System is still used to number chapters and sentences in the bible. It uses combinations of Hebrew letters to represent numbers. Using the English alphabet for simplicity it works like this. A is 1, B is 2, ... , I is 9, J is 10, K is 20, ... R is 90, S is 100, T is 200, U is 300, V is 400. (There are only 22 letters in the hebrew alphabet). Numbers that are not listed above are created by using a unique combination of letters, where each letter used from left to right is the largest number one can use without getting too high. For example, 32 would be written as LB; 156 would be SNF. The only exceptions are 15 and 16 which normally would be written JE and JF respectively, but since the corresponding Hebrew letter combinations spell God's name, the numbers are instead represented by IF and IG. Normally, numbers greater than 400 do not show up much, but any number can be represented by using an appropriate number of V's at the start. For example, 1000 would be VVT.

a. Is this system positional, symbolic, both, or neither? Why?

b. Using the English alphabet A-V, how would you write the numbers 716, 98, and 499?

c. What is the largest number you can represent without using any double V's.

2. More Nim (10 points)

In another version of Nim, you start with an odd number of sticks where each player may take 1, 2 or 3 sticks on each turn, and the goal is to accumulate an odd number of sticks after all the stick shave been taken.

Analyze this game and determine if there is any kernel. Conjecture what conditions determine whether you will go first or not, and prove your conjecture.

3. A Medieval Puzzle (10 points)

In the 1200's Levi Ben Avraham, wrote a book with 10 chapters, each on a different topic including ethics, Logic, Creation, The Soul, Prophecy, The Merkabah, Mathematics, Astronomy and Astrology, Physics, Metaphysics. Levi himself is not considered a great scholar although he did some original scientific work.

In the chapter on Mathematics, he poses the following "question" or problem with the solution.

Question: A man travel 15 miles each day. Another man travels 1 mile on the first day, 2 miles on the second day, and so on, adding a mile each day. After how many days will they meet? Answer: Add the number of days that they did not walk the same number of miles, which is 14, to the number of the day on which they did walk the same number of miles, which is 15, and you get 29.

a. Generalize this method to show the solution for n, where n is the number of miles traveled in a day by the first man.

b. Prove that your method works.

4. Aristotle's Proof Revisited (25 points)

a. Explain where the proof below is incorrect. It tries to prove that the sqaure root of 4 cannot be written as a fraction a/b. Note the square root of 4 is 2/1.

Assume that &Atilde4 = a/b, where without loss of generality, a and b are in lowest terms.

Then 4 = (a*a)/(b*b), and hence 4*b*b = a*a.

Therefore, a*a is divisible by 4. If so, then a must be divisible by 4, let a =4*k.

Then 4*b*b = 4*k*4*k, and b*b = 4*k*k, so b is divisible by 4.

But if a and b are both divisible by 4, then a/b is not in lowest terms!!

b. Mimick Aristotle's proof to prove that &Atilde3 cannot be written as a/b, and provide all the details to completethe proof correctly. (Look back at the lemmas in the homework).

5. Levi's Square Roots (25 points)

In 1321, Rabbi Levi ben Gershon (grandson of Levi ben Avraham), described a method to extract square roots of perfect squares. The example he gives is translated below.

For example, if you want to take the square root of 82646281, look at 82, and note that 81 is the closest square and its root is 9. Then write 9 down in the "answer" row under the 4th column from the right, which is the middle column between 82 and the right. Then subtract 81 from 82 leaving 1646281. Now, the current root is 9, and 2*9=18 does not divide into 1 so bring down the next digit, namely 6. But 18 does not divide 16, so bring down the next digit, namely 4. Now 18 does divide 164, and the result is 9 which should be written in the "answer" row in the second column from the right. Now multiply twice the current root (2*9000) by this new number (90) and add in the new number muliplied by itself (90*90), then subtract all this from 1646281, leaving 18181. Now the current root is 909, and 2*909=1818 divides 1818 and the result is 1, so put 1 in the rightmost column. Now multiply twice the current root (2*9090) by this new number (1) and add in the new number multiplied by itself, then subtract all this from 18181 and nothing is left so you are done. And the square root of 82646281 is 9091.

a. Show how to use Levi's method on 2182010944.

b. Generalize Levi's method for numbers that are not perfect squares. Show how to use this method on 1234567 to find the square root to the nearest 100th.

c. Levi explains the reason behind his method with the following brief sentence.

This all works because as we explained before, when you add one number to another and square the sum, the result is equal to the squares of each number plus twice the product of the two numbers.

Interpret his explanation and explain in your own words what is going on.