Mathematics Problems, Puzzles, and Challenges

All of the problems below are of my own devising. I've tried to include problems of varying levels of difficulty and from diverse areas within mathematics. Comments are welcome.

Note: Whenever a problem calls for a numerical solution, try to give an __exact__ answer (i.e., in terms of elementary functions, radicals, standard numerical constants, etc.), as opposed to a decimal approximation, if possible. Unless specified otherwise, assume all numbers are in base 10 and are written out in ordinary standard notation.

1. Evaluate the following expression:

2. An "algebraic number" is defined as a number that is a zero of a polynomial function with integer coefficients. Thus the square root of 2 is algebraic, since *f*(*x*) = *x*^{2}–2 evaluates to 0 if the square root of 2 is substituted for *x*. Similarly, the cube root of 2 is algebraic, since it is a zero of *f*(*x*) = *x*^{3}–2. A theorem states that the sum of algebraic numbers must be an algebraic number. According to this theorem, _{}is algebraic. Find a polynomial function with integer coefficients that has as a zero.

^{
}3. Let *s* be the sum of the units digits of the *n*th powers of 13 consecutive positive integers, the smallest of which is *k*. (*n* must also be a positive integer.) Find the minimum value for *s*; the minimum value for *n* that yields that value for *s*; and the minimum value for *k* that gives those values for *s* and *n*.

4. Evaluate the following expression:

5. Evaluate the following expression:

_{
}

6. Evaluate the following expression:

_{
}

7. Skewes' Number, in its common form (using 10 as a base), equals

_{ }^{.}

How many distinct factors does *the square root of Skewes' Number_{ }*have?

8. How many distinct (i.e., mutually non-congruent) triangles are there such that, for

xxx• the side-lengths are integers in geometric progression

xxx• at least one side has length 100 ?

9. How many distinct (i.e., mutually non-congruent) triangles are there such that, for

xxx• the side-lengths are integers in arithmetic progression

xxx• at least one side has length 100 ?

The following two problems, and some subsequent ones, require the use of standard English words for the names of the counting numbers: "one," "two," "three," and so on. Please use the American system (e.g., 10

10. What number's name is first in alphabetical order among the perfect squares (in the specified set)?

11. What number's name is last in alphabetical order among the perfect squares (in the specified set)?

12. This one is basically a tedious exercise in silliness, but I'm including it anyway because it exhibits one of the most astonishing coincidences I've ever encountered. . . . .

Make a list of the names of the first 100

Next, alphabetize this list. (I told you it was tedious!) Your new list should begin with "eighteenth," "eighth," "eightieth," and end with "twenty-seventh," "twenty-sixth," "twenty-third."

Now, down the left margin of a sheet of ruled paper, write the 26 letters of the alphabet (in order), one per line.

Next to the letters, write the first 26 alphabetized number names in a column ("eighteenth" next to "A"; "eighth" next to "B"; and so on).

Then start a new column with the

Then start another new column with the

(There aren't enough number names remaining to make yet another complete column of 26, so don't bother.)

What number name concludes the

What number name concludes the

What number name concludes the

13. Define

xxxxx

xxxxx

Evaluate

14. What are the coordinates of the incenter of a triangle whose vertices are at (3, 1), (5, 7), and (10, 3)? (Use ordinary rectangular coordinates.)

15. Solve each of the following equations for

a)

b)

c)

d)

16. Begin a sequence (

17. A type of challenge frequently seen in mathematical puzzle compendia requires the solver to construct expressions equal to various numbers, using a "tool kit" consisting of standard math symbols and some specified small number of repetitions of a particular digit. For example, I've seen (and tried my hand at) a challenge the object of which is to construct positive integers from four 4s and the symbols for addition, subtraction (unary "negative sign" OK), multiplication, division, factorial, and square root; as well as the decimal point and parentheses (for associative inclusion only [no combinatorics!]). Exponentiation and use of place value (base 10) are also allowed. Among those symbols and operations NOT allowed: combinations and permutations, subfactorial, greatest integer function, absolute value, bar or ellipsis for repeating decimal digits, radical symbol for roots other than square roots, conversion to non-decimal bases. Examples for the first few positive integers might be:

1 = (4/4)×(4/4)

2 = (4/4)+(4/4)

3 = (4+4+4)/4 , and so on.

a) Using only those symbols and operations specified above, and

b) Using those same symbols and operations, and still

c) Using the same set of symbols and

d) Using the same set of symbols and

18.

19. What is the arithmetic mean of the set of all (nonzero) perfect squares less than or equal to one million? What is the median of this set?

20. (

a) a positive number ending in 1? (i.e., the units digit is 1)

b) a negative number ending in 1?

c) a positive number ending in 2?

d) a negative number ending in 2?

e) a positive number ending in 3?

f) a negative number ending in 3?

g) a positive number ending in 4?

h) a negative number ending in 4?

i) a positive number ending in 5?

j) a negative number ending in 5?

k) a positive number ending in 6?

l) a negative number ending in 6?

m) a positive number ending in 7?

n) a negative number ending in 7?

o) a positive number ending in 8?

p) a negative number ending in 8?

q) a positive number ending in 9?

r) a negative number ending in 9?

s) a positive number ending in 0?

t) a negative number ending in 0?

u) equal to 0?

21. Let the "degrino" be a unit of angular measure that divides a circle into 20! (20 factorial) equal parts. For how many

"The degrino measure of each interior angle of a regular

22. Find the side-lengths of the triangle of minimum perimeter such that each side-length is a perfect square, and also the triangle is:

a) (no additional restrictions)

b) isosceles but not equilateral

c) scalene

d) right

e) acute

f) obtuse

g) obtuse and scalene

h) acute and scalene

23. How many cubic revo-Planck lengths are there in 1 cubic una-Hubble length?

24. In establishing the countability of the set of algebraic numbers (see #2 above), the mathematician Georg Cantor used a concept known as the "height" (sometimes called "weight") of a polynomial. Let us define the height of a polynomial

"The height of a polynomial with integer coefficients is defined to be

the degree of the polynomial + the sum of the absolute values of the integer coefficients – 1 ."

Example: The height of

How many distinct polynomial expressions in

25. A tire commercial claimed: "1 out of every 4 vehicles on the road has at least one underinflated tire." Suppose this is true. For this problem, assume each vehicle has exactly 4 tires. Assume also that underinflation is a random event whose probability is the same for all tires. What proportion of all tires (on vehicles on the road) is underinflated?

26. Suppose you are at

27. How many distinct (i.e., mutually non-congruent) triangles are there such that, for

xxx• the degree measure of each angle is a perfect square

xxx• at least one side-length is a perfect square

xxx• the perimeter is less than 1000 ?

28. Find a polynomial function

29. I finished high school in 1974 and graduate school in 1979. In honor of these dates, find the distance between the parallel lines with equations

*y* = _{}*x *+74

and

*y* = _{}*x *+79 .

30. a) How many distinct (i.e., mutually non-congruent) triangles are there such that, for *each* triangle, all of the following are true:

xxx• the radian measures of at least two angles are integers

xxx• the perimeter is a perfect square

xxx• the perimeter is less than 1000 ?

......b) Of the set of triangles satisfying the conditions specified in part a), consider the one with maximum area. What is its perimeter? Its angle-measures? Its side-lengths? Its area?

31. In a right triangle of area 1, the hypotenuse is 1 unit longer than one of the legs. What is the perimeter of the triangle?

32. An equilateral triangle has area 1. A circle (in the same plane) whose center is coincident with the triangle's circumcenter intersects the triangle in such a way that all its arcs whose endpoints are consecutive points of intersection between the circle and triangle are congruent. It is neither the inscribed circle nor the circumscribed circle of the triangle. What is the radius of this circle?

33. A square has area 1. A circle (in the same plane) is concentric with the square and intersects it in such a way that all its arcs whose endpoints are consecutive points of intersection between the circle and square are congruent. It is neither the inscribed circle nor the circumscribed circle of the square. What is the radius of this circle?

34. (For this problem, "log" and "logarithm" refer to common [base-10] logarithms; "average" refers to arithmetic mean.)

I once ran a computer program to calculate the average logarithm of all 1-digit (positive) integers; of all 2-digit integers; and so forth. The calculations indicated an average log for 1-digit integers of approximately 0.61775; the average was 1.67122 for 2-digit integers, 2.67626 for 3-digit integers, and 3.67676 for 4-digit integers. I conjectured that the mantissas (the quantities after the decimal points) might be approaching a limit. Some subsequent analysis showed this to be true.

Let *m _{n}* represent the average log of all

What is

36. Consider the function

37. a) What is the smallest positive integer that has exactly 100 distinct factors?

xxxb) What is the smallest positive integer that has more than 100 distinct factors?

38. A four-digit (base-10) integer is chosen at random. What is the probability there will be more 1s than 2s among its digits?

39. This question, like Problems 10, 11, and 12 above, involves the names in the American system for the positive integers less than 10

xxxa) In an alphabetical list of the names in the American system of the first 10

xxxb) What position on such a list of alphabetized ordinals is occupied by "eighth"?

40. a) For how many integral values of

_{} is an integer.

41. What is the smallest positive integer that includes among its factors at least 10 perfect squares?

42. a) How many distinct (i.e., mutually non-congruent) triangles are there such that, for

xxx• at least two sides are congruent, and the length of each of the congruent sides is a 1-digit integer

xxx• the area is an integer ?

......b) Of the set of triangles satisfying the conditions specified in part a), consider the one(s) with maximum area. Give the perimeters of all such triangles.

43. What is the area (to the nearest square unit) of the violin on my home page? (It is bounded by

44. How many

45. A real number

46. Consider the set of all quantities

47. The degree measures of the angles of an acute triangle are all prime numbers. One side—not the shortest—has length 1. What is the area of the triangle?

48. a) How many different combinations of three integers could represent the degree measures of the angles of a triangle?

b) In how many of these combinations are all three integers prime?

49. The maximum possible number of angles of a scalene triangle have degree measures that are perfect numbers. What is the minimum possible area of the triangle if the perimeter is also a perfect number?

50. Mathematician A says, "I'm thinking of an integer

51.

a) All the edges of a regular pentagonal pyramid have length 1. What is the volume of the pyramid?

b) All the edges of a regular pentagonal pyramid are congruent. The volume of the pyramid is 1. What is the length of each edge?

52. Give all values of

53. Consider the names of the integers between 1 and 1,000,000, inclusive. Of these:

a) how many are spelled with more consonants than vowels?

b) how many are spelled with an equal number of consonants and vowels?

c) how many are spelled with more vowels than consonants?

54. Consider the set of names of the integers between 1 and 10

a) What is the minimum value of C–V?

b) How many members of the set have this value?

55.

a) Give the ranges of possible values for

b) Same as a), but with the added condition that at least one of

c) Same as a), but with the added condition that at least two of

d) Same as a), but with the added condition that

56. Find

57.

58. Five integers are chosen at random. What is the probability that

1) One or more of the integers is a multiple of 10.

2) The members of at least one pair of those integers differ by a multiple of 10.

59. Define round(

60. Consider the standard names (in American English) of the positive integers that are between 1 and 10

a) What number's name is first in alphabetical order among the factorials (in the specified set)?

b) What number's name is last in alphabetical order among the factorials (in the specified set)?

61. Consider the standard names (in American English) of the positive integers that are between 1 and 10

a) What number's name is first in alphabetical order (in the specified set)?

b) What number's name is last in alphabetical order (in the specified set)?

62. How many distinct (i.e., mutually non-congruent) triangles are there such that, for

• the radian measures of at least two angles are integers

• the length of at least one side is an integer

• the perimeter is less than 100 ?

63. How many five-digit integers (in base 10) are powers of smaller integers?

64. On an ordinary set of x–y coordinate axes, place a unit circle with its center at the origin. Find the coordinates of a point (

65. Let

a) What is the minimum possible value of

b) What is the maximum possible value of

c) What is the maximum possible value of

d) For what value of

e) For what value of

f) What is the minimum possible value of

g) Give a formula for

h) What is the maximum possible value of

66. Consider the set of all quantities

67. a) In the decimal expansion of π, how many factors does the product of the first 10 digits after the decimal point have?

b) How many factors does the product of the first 100 digits after the decimal point have?

68. Consider the (infinitely) repeating decimal 0.123456789 123456789 123456789 ... .

a) How many factors does the product of the first 1000 digits after the decimal point have?

b) How many factors does the

69. Four integers are chosen at random. What is the probability their product will end in 0?

70. Let A be the sequence whose

71. Let

a) What is the smallest

b) For how many different values of

72. Suppose you have an ordinary old-fashioned nondigital clock (with the numbers 1 through 12 spaced evenly around the circumference of a circular face [12 at the top, if you like], and an hour hand and a minute hand that rotate continuously at a constant rate). For this problem, let the time always be expressed as a whole number of hours and a precise number of minutes. (Example: At 2:05 and 15 seconds, the time would be given as 2 hours,

a) When would be the first time after 12:00 that the measure, in degrees, of the smaller angle between the hour hand and the minute hand would equal the sum of the hour and the minutes past the hour? When would be the second time? The third?

b) When would be the first time after 12:00 that the measure, in degrees, of the smaller angle between the hour hand and the minute hand would equal the

73. How many different multisets of positive integers can be constructed each having the property that 36 is both the sum and the product of all its elements? (A multiset is just a set where repeated elements are allowed.) List them.

74. Each of

a) 1?

b) 2?

c) 3?

d) 4?

e) 5?

f) 6?

g) 7?

h) 8?

i) 9?

j) 0?

75. For how many positive integers less than or equal to 1,000,000 is the product of the digits a power of 8? (Note: 8

76. a) What is the smallest positive integer the product of whose digits is (exactly) a googol?

b) What is the smallest positive integer the product of whose digits is greater than a googol?

77. a) What is the smallest positive integer such that the product of its digits equals 1,000,000, and no digit appears twice in a row?

b) The largest?

78. What is the smallest positive integer such that the sum of its digits equals 100, and no digit appears twice in a row?

79. How many elements are in the set that consists of every

80. a) For how many integers between 1 and a googol (inclusive) is the product of the digits a prime number?

b) For how many integers between 1 and a googol (inclusive) is the product of the digits a one-digit positive number?

81. a) What is the smallest positive integer for which the product of its digits is exactly 100 times the sum of its digits?

b) What is the smallest positive integer for which the product of its digits is more than 100 times the sum of its digits?

82. How many positive integers have 100 as both the sum and the product of their digits?

83. Find the smallest positive integer

a) at least one 0.

b) at least one 1.

84. Find the smallest positive integer

85. Consider the set of positive integers for which there are, among the digits of each:

• more 0s than 1s;

• more 1s than 2s;

• more 2s than 3s;

• more 3s than 4s;

• more 4s than 5s;

• more 5s than 6s;

• more 6s than 7s;

• more 7s than 8s; and

• more 8s than 9s.

a) What is the smallest such integer?

b) What is the largest such integer that is less than a googol?

86. Let

87. Find the longest sequence of consecutive integers such that the product of their factorials is a factor of 100!.

88. In the decimal expansion of π, what is the sum of the first 500,000 digits after the decimal point? Express your answer in scientific notation, accurate to three significant digits.

89. What is the sum of the digits of the number that is equal to 10

90. Find the smallest integer

a)

b)

91. Find the smallest integer that is divisible by:

a) every positive even integer less than 100.

b) every positive odd integer less than 100.

92. What is the smallest four-digit prime number whose digits alternate in parity (i.e., being even or odd)?

93. a) How many six-digit positive integers have the property that the sum of their digits is greater than the product of their digits?

b) Among those, how many do not contain the digit "0"?

94. Consider those prime numbers that have more even digits than odd digits.

a) What is the smallest?

b) The second smallest?

c) The largest that is less than 1,000,000?

95. a) What is the largest integer not containing the digit "1" for which the product of its digits equals the factorial of any integer?

b) Let

96. a) Let P(

b) Let

c) For how many different values of

97. An integer between 1 and 1,000,000 (inclusive) is chosen at random, and its digits are multiplied together. Give, as an exact percentage, the probability that the product will be:

a) an odd positive number;

b) an even positive number;

c) neither.

98. Define a sequence A as follows:

A(1) = 1 ;

A(

a) What is A(100) ?

b) Find a closed-form formula for A(

99. Begin with any integer

a) How many steps will it take to reach 1, as a function of

b) How many digits are in the smallest

100. An integer between 1 and 1,000,000 (inclusive) is chosen at random. Give, as an exact percentage, the probability that it will have:

a) more even digits than odd digits;

b) more odd digits than even digits;

c) an equal number of even and odd digits.

101. An integer between 1 and 1,000,000 (inclusive) is chosen at random. Give, as an exact percentage, the probability that the product of its digits will be:

a) a multiple of 6, if 0 counts as a multiple;

b) a positive multiple of 6;

c) not a multiple of 6.

102. For how many integers between 1 and 1,000,000 is the sum of their digits at least 50?

103. How many integers between 1 and 1,000,000 have the property that both the sum and product of their digits are prime numbers? List them.

104. What is the sum of the last 25 digits of the 158-digit number that equals 100! ?

105. Construct the set whose elements are those integers between 1 and 1,000,000 for which the product of their digits equals 1,000.

a) How many elements are in the set?

b) What is the median of the set?

c) What is the expected sum of the digits of a randomly chosen element of the set?

d) If a single digit from one of the elements of the set is chosen at random, what is its expected value?

106. The circumference of a unit circle is 2π, or approximately 6.2831853072. The perimeter of a polygon inscribed in a unit circle will be less than 2π, but as the number of sides increases the perimeter will approach that value. What is the minimum

a) 6?

b) 6.2?

c) 6.28?

d) 6.283?

e) 6.2831?

f) 6.28318?

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