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) = x2-2 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) = x3-2. 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 nth 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 each triangle, both of the following are true:
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 each triangle, both of the following are true:
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., 106 = one million, 109 = one billion, 1012 = one trillion), not the British system. Example: 1,234,567,890,987,654,321 would be read as "one quintillion, two hundred thirty-four quadrillion, five hundred sixty-seven trillion, eight hundred ninety billion, nine hundred eighty-seven million, six hundred fifty-four thousand, three hundred twenty-one." For these problems, consider the names of only those positive integers that are between 1 and 1066–1, inclusive.

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 ordinal numbers. Your list should begin with "first," "second," "third," and end with "ninety-eighth," "ninety-ninth," "one hundredth."
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 next 26 alphabetized number names ("fortieth" next to "A"; "forty-eighth" next to "B"; and so on).
Then start another new column with the next 26 alphabetized number names ("seventeenth" next to "A"; "seventh" next to "B"; and so on).
(There aren't enough number names remaining to make yet another complete column of 26, so don't bother.)
What number name concludes the first column?
What number name concludes the second column?
What number name concludes the third column?

13. Define n? for each positive integer n as follows:
xxxxxn? = 1+3+5+...+n, if n is odd;
xxxxxn? = 2+4+6+...+n, if n is even.
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 x (over the real numbers) so that the equation will be true for every integral n:
a)

b)

c)

d)

16. Begin a sequence (An) with a1 = 1; a2 = 1; a3 =1; then generate each subsequent term by taking the sum of the previous three terms (i.e., this is similar to the Fibonacci sequence except that three terms instead of two are added together). As the number of terms increases, the ratio between each term and its predecessor approaches a limit. Find this limit. I.e., find

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 only TWO 4s, how many different integers between 0 and one googol, inclusive, can you construct? (I have found 62.)
b) Using those same symbols and operations, and still only TWO 4s, how many different non-integral rational numbers between .01 and 100 can you construct? (I have found 28.)
c) Using the same set of symbols and THREE 4s, find an expression that equals (the mathematical constant) pi.
d) Using the same set of symbols and FOUR 1s, find an expression that equals one quattuordecillion.

18. (Problem deleted)

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. (REVISED!) Suppose you perform an experiment as follows: roll an ordinary fair six-sided die and record the number k that comes up. Choose a random element x from the set whose members are all those numbers that are the product of k consecutive integers. What is the probability (over the entire experiment) that x is:
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 different values of n is the following statement true?
"The degrino measure of each interior angle of a regular n-gon is an integer."

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:
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 with integer coefficients as follows:
"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 x2-3x-1 = 2+(1+3+1)-1 = 6 .
How many distinct polynomial expressions in x are there of height 10? (Rearrangement of terms does not count as distinct!)

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 P on the earth's surface, and your friend is at Q. Points P and Q are both in the Northern Hemisphere, and also both are in the Eastern Hemisphere. The east longitude of P is three times the north latitude of P. The north latitude of P is twice that of Q; the east longitude of Q is twice that of P. The geodesic (i.e., great-circle) distance between P and Q is 1500 miles. Assume the earth is a smooth sphere of radius 3963 miles. How would you and your friend exchange greetings in the native tongues of your respective localities?

27. How many distinct (i.e., mutually non-congruent) triangles are there such that, for each triangle, all of the following are true:
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 f such that the tangent lines to the graph of f at (1, f(1)), (2, f(2)), (3, f(3)), and (4, f(4)) enclose a square whose area is the arithmetic mean of f(1), f(2), f(3), and f(4).

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 mn represent the average log of all n-digit integers (for positive integral n).
What is ?

35. Line L1 has the equation y = mx+1. Line L2 has the equation y = mx+2 (same m as for L1). m is a positive number such that the region within the second quadrant that is between L1 and L2 has area 1. What is the perimeter of this region?

36. Consider the function S that associates with each positive integer n the sequence containing the numbers of factors (divisors) of each of the positive integers between 1 and n, inclusive. For example, S(6) would be (1, 2, 2, 3, 2, 4). What is the smallest value of n for which S(n) has a single mode, and that mode is greater than 2?

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 1066. (The principles of construction of these names can also be found in large dictionaries [e.g., Webster's Third New International Dictionary at the entry for "number"].) If the set of names of the first 1066–1 cardinal numbers (one, two, three, etc.) is alphabetized, the list will begin with "eight." But if the ordinal forms (first, second, third, etc.) are alphabetized, "eighth" will not be the first entry.
xxxa) In an alphabetical list of the names in the American system of the first 1066–1 ordinal integers, what name will be first?
xxxb) What position on such a list of alphabetized ordinals is occupied by "eighth"?

40. a) For how many integral values of n in the range 1 to 1000 (inclusive) is the following statement true?

is an integer.

b) (NEW!) Among all the integer values for the expression shown in part a) (i.e., those that correspond to values of n that satisfy the conditions stated), which one has the most factors? How many?

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 each triangle, both of the following are true:
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 x = –32 and x = 32.)

44. How many n-digit positive integers include exactly two 1s among their digits?

45. A real number x that is (strictly) between 0 and 10 is chosen at random. Let x' be the number that results when the integer part of x is multiplied by 10 but the fractional part remains the same. (Example: If x is 3.462, then x' would be 30.462.) What is the expected value of x'/x?

46. Consider the set of all quantities p/q such that p and q are integers, 0 < p < q < 1,000,000, and the decimal representation of p/q terminates. What is the maximum of this set?

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 k. I'll show you only the rightmost digit of the product of k × (k+6), and you should be able to tell me what k is." Mathematician B responds, "You don't need to show me anything—I already know." What is k?

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

52. Give all values of n for which the following statement could be true:

The degree measures of the angles of an n-gon are consecutive integers.

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 1066–1, inclusive. For each member of the set, let C = the number of consonants and V = the number of vowels.
a) What is the minimum value of C–V?
b) How many members of the set have this value?

55. x, y, and z are positive real numbers. x < y < z, and x+y+z = 10.
a) Give the ranges of possible values for x, for y, and for z.
b) Same as a), but with the added condition that at least one of x, y, or z must be an integer.
c) Same as a), but with the added condition that at least two of x, y, and z must be integers.
d) Same as a), but with the added condition that x, y, and z must be integers.

56. Find .

57. k distinct positive numbers, of which exactly j are integers, sum to 10. Describe the set of all possible ordered pairs (j, k).

58. Five integers are chosen at random. What is the probability that at least one of the following is true?
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(x) as the closest integer to x.* Find the total length of all intervals of x values in which round(x2) and round(x) are both positive numbers that have the same number of digits.
*Solving this problem doesn't require specifying whether half-integers are rounded up or down, but if you wish to make the definition unambiguous you are welcome to choose one or the other!

60. Consider the standard names (in American English) of the positive integers that are between 1 and 1066–1.
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 1066–1, . . . spelled backwards.
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 each triangle, all of the following are true:
• 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 (p, q) on the circle such that the tangent to the circle at (p, q) contains a segment of length 5 that lies in the first quadrant and whose endpoints are on the axes.

65. Let S be the sum of the degree measures of all the interior angles of an n-sided polygon.
a) What is the minimum possible value of n for which the sum of the digits of S is at least 100?
b) What is the maximum possible value of n for which the sum of the digits of S is less than 10, and no digit of S other than the units digit is 0?
c) What is the maximum possible value of n for which the digits of S are all distinct?
d) For what value of n would the name of S be the first in alphabetical order among all possible values of S when spelled out in English?
e) For what value of n would the name of S be the last in alphabetical order among all possible values of S when spelled out in English?
f) What is the minimum possible value of n for which the name of S, when spelled out in English, contains the letter "L"?
g) Give a formula for n in terms of k such that the corresponding value of S will be the kth smallest perfect square among all possible values of S.
h) What is the maximum possible value of n for which the digits of S, read left to right, are strictly decreasing?

66. Consider the set of all quantities p/q such that p and q are integers, and 0 < p < q < 1,000,000. If the maximum of this set is represented in decimal form, how many factors does the product of the first 100 digits after the decimal point have?

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 sum of the first 1000 digits after the decimal point have?

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

70. Let A be the sequence whose nth element is the smallest n-digit positive integer the product of whose digits is an n-digit integer. List the entire sequence A.

71. Let f be the function on nonnegative integers such that f(n) is the base-10 number that results when each digit of n is replaced by its equivalent in binary. (Example: f(27) = 10,111.)
a) What is the smallest n for which f(n) exceeds a googol?
b) For how many different values of n does f(n) = 11,111? What is the smallest such n?

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, 5.25 minutes.)
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 product of the hour and the minutes past the hour? When would be the last time before 12:00 that the measure, in degrees, of the smaller angle between the hour hand and the minute hand would equal the product of the hour and the minutes past the hour?

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 and b is a randomly selected positive integer. What is the probability that the units digit of ab is:
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: 80 counts as a power.)

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 distinct quantity that is the average (arithmetic mean) of the digits of an integer between 1 and 1,000,000, inclusive?

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 N such that if a positive integer k that is less than or equal to N is selected at random, the probability is greater than not that the digits of k will include:
a) at least one 0.
b) at least one 1.

84. Find the smallest positive integer N such that if a positive integer k that is less than or equal to N is selected at random, the probability that all the digits of k will be the same is greater than the probability they will all be different.

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 P be the product of all the distinct factors of 100! (100 factorial). P has d digits, the last k of which are 0s. What is d, and what is the maximum value of k?

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 10100–1099–1098 – . . . –101–100 ?

Solutions