PROBLEMS FROM THE MATHEMATICS TEACHER

In the early 1980s, the NCTM's journal for secondary mathematics teachers, The Mathematics Teacher, began publishing a monthly calendar featuring interesting facts and problems for every day of the month. The problems listed below are a selection which seemed to me to be particularly suited for solving through the use of spreadsheet. These are mostly of the nature of finding integers which fit a variety of conditions. I excluded problems which depended more on following a given algorithm and selected problems that required searching through a range of numbers for a good fit. Also, problems which asked to find a particular digit within a number were excluded because of the complexity of fitting this to a spreadsheet application. Word problems and logical exercises were excluded because they depended more on intuitive thought and a limited number of steps which a spreadsheet could not really assist with.

Undoubtedly, there are many more problems from The Mathematics Teacher that could be solved using a spreadsheet, with a bit of ingenuity. These stood out to me as being obviously benefitted by the use of spreadsheets, or problems that would be easily extended and broadened by the same. There are a total of 58 problems included here, many of which are similar in nature and could be grouped as a unit.

I. 1983, Vol. 76

A. September, No. 6

23- Multiply any four consecutive positive integers and add 1 to the product. Will you always get a perfect square? Prove your answer.

B. October, No. 7

10- Which is larger, 2¹ or 1²? 3² or 2³? 4³ or 3⁴? 7⁶ or 6⁷? ... ? Can you predict the larger of any two positive integers that follow this pattern?

12- If a faucet drips at the rate of a drop every 2 seconds and it takes 575 drops to fill a 100-mL bottle, how many liters of water are wasted in a week? A month? A year?

C. November, No.8

21- A theater had 100 people in the audience consisting of men, women, and children. The prices of the tickets were $10.00-men, $3.00-women, and $0.50-children. The theater collected a total of $100.00 from 100 people. How many men, women, and children were in the audience?

27- Find three integers in arithmetic progression whose product is a prime.

II. 1984, Vol. 77

A. April, No. 4

7- The Fibonacci sequence is 1,1,2,3,5,8,13,21,... Pick any four adjacent numbers in the sequence. How does the product of the inner two numbers compare with the product of the outer two?

8- The smaller of two consecutive natural numbers is divisible by 23 and the larger by 29. Find the smallest pair of such numbers with the property that they both contain only the digits 1 and 2.

16- Pick any five numbers in the Fibonacci sequence. How does the square of the middle number compare with the product of the first and the last terms?

17- Find six consecutive positive integers that add to 87.

22- Find three consecutive odd numbers the sum of whose squares consists of four identical digits.

28- Find all pairs of unequal positive integers that have the property that the difference of their squares is five times the difference of the numbers themselves.

B. May, No. 5

3- About how many times will your heart beat during this month?

23- What three consecutive whole numbers have a product of 29,760?

24- Which pairs of two-digit consecutive numbers have squares that differ by a perfect square?

28- Choose any number, multiply by 2, add 5, multiply by 5, subtract 25, and divide by 10. Compare your result to the original number.

29- Choose any number, multiply by 3, add 8, add your original number, divide by 4, and subtract your original number. Compare your result to the original number. Write an equation using n for some number that describes the steps in this problem and in the problem of 28 May.

III. 1985, Vol. 78

A. December, No. 9

1- What is the smallest number divisible by the numbers 1 through 9?

16- What is the sum of 1*1!+2*2!+3*3!+ ... +n*n!?

IV. 1986, Vol. 79

A. May, No. 5

6- If it is 11:11 a.m., what time will it be 143 999 999 993 minutes later?

B. September, No. 6

2- Find the sum: .

21- Find the greatest odd factor of 7992.

28- What is the largest three-digit number divisible by both 9 and 7?

V. 1987, Vol. 80

A. January, No. 1

2- Darryl ate 100 peanut butter cups in five days. Each day he ate six more than he ate the previous day. How many peanut butter cups did Darryl eat on the first day?

8- You have the same number of nickles and quarters. The total value is $3.30. Find the number of quarters.

11- Solve the magic square:

42		40
38	43

27- If ab = (a+b)+(ab)+b, what is (57) 3)?

B. December, No. 9

9- Show that by adding one to the product of four consecutive integers a square is obtained.

25- Find all possible sets of four consecutive integers such that the sum of the cubes of the smallest three is the cube of the fourth.

VI. 1990, Vol. 83

A. September, No.6

3- How many nonprime natural numbers are factors of 8575?

7- Find the difference between the sum of all even numbers and the sum of all odd numbers from 0 through 1000.

26- At one of George Washington's parties, each man shook hands with everyone except his spouse, and no handshakes took place between women. If thirteen married couples attended, how many handshakes were there among these twenty-six people?

28- The increasing sequence 2, 3, 5, 6, 7, 10, 11, ... consists of all positive integers that are neith the square nor the cube of a positive integer. Find the 500th term of this sequence.

30- Let n be the smallest positive integer that is a multiple of 75 and has exactly 75 positive integral divisors, including 1 and itself. Find n/75.

VII. 1991, Vol. 84

A. January, No. 1

1- When five consecutive integers are added, their sum is 155. Find the integers.

3- Solve for x to the nearest thousandth in x⁵-x-3 = 0.

B. February, No. 2

7- In a special game of football, a touchdown is counted as 8 points with a possibility of adding 2 points after a touchdown is mad; a field goal counts 5 points. What number less than 35 represents the largest score that cannot be made in this game?

9- For what values of n is a perfect square?

25- Determine all possible values of n such that (1-n)³ + 3(1-n)²*(n) + 3(1-n)n²+n³ = 0.

C. March, No. 3

1- If y^*= y²-1, find (y^*)^*.

6- Find two numbers such that their difference, sum, and product are in the ratio of 1:4:15, respectively.

13- What is the units digit in (1988)¹⁹⁸⁷?

20- For every positive integer n, n³-n must be divisible by what positive integers?

22- How many four-digit numbers abcd exist such that a is even, b is divisible by 5, c is a prime, and d is odd?

D. April, No. 4

9- Given the following sequence of fractions:

1/1;

1/2, 2/2, 1/2;

1/3, 2/3, 3/3, 2/3, 1/3;

1/4, 2/4, 3/4, 4/4, 3/4, 2/4, 1/4; ...

What is the position of 7/10 in the sequence?

What fraction is in the 400th position?

26- A square sheet of paper measuring x units on a side has a 1-unit-square hole in it. The remaining paper area is x square units. How long is the side of the paper?

E. May, No. 5

19- Find a two-digit number, the sum of whose digits is equal to the square of its cube root.

F. September, No. 6

8- The sum of a whole number and the next four consecutive whole numbers is 105. Find the result when the mean of the numbers is subtracted from the median of the numbers.

15- Fiven f(x)= -|x-4| and g(x) = (1/2)x²-8, for how many integral values of x is g(x) less than f(x)?

G. October, No. 7

29- What is the largest product that can be obtained from a sum of positive integers that total 100?

H. November, No. 8

2- After a committe meeting where ten people sat around a circular table, each person shook hands with everyone else except the people who sat on either side. How many handshakes took place?

7- Pythagoras discovered amicable, or "friendly" numbers. Two positive integers are amicable if each is the sum of the proper divisors of the other; 284 and 220 are amicable. What number is amicable to 1184?

8- Solve this problem from Problems for the Quickening of the Mind, probably compiled by Alcuin of York (ca.775): If 100 bushels of corn is ... distributed amon 100 people [in such a manner] that each man receives 3 bushels, each woman 2 and each child 1/2 of a bushel, how many [men, women, and ] children are there?

14- A classic problem from Alcuin of York: A dog chasing a rabbit which has a [head start of 150 feet] jumps 9 feet every time the rabbit jumps 7. In how many leaps [does] the dog overtake the rabbit?

19- The number 2¹⁶-1 is divisible by four prime numbers. Find these numbers.

30- How many numbers less than 124 are divisible by 2, 3, and 5?

I. December, No. 9

6- My license tag is a three-digit number. The product of the digits is 216, their sum is 19, and the digits appear in ascending order. Find the license-plate number.

12- On a scavenger hunt, you are hunting for two integers that satisfy the following conditions: (a) both integers are even, (b) the sum of the integers is negative, (c) the product of the integers is -96, (d) neither integer is a square, (e) neither integer is a factor of the other, and (f) neither integer is a cube. What are the two integers?

14- Compute: .