# Problem of the week, 161

A former colleague Bill Wardlaw (March 3, 1936-January 2, 2013) used to create a “Problem of the Week” for his US Naval Academy students, giving a prize of a cookie if they could solve it. One of them is given below.

The residue of an integer n modulo an integer d > 1 is the remainder r left when n is divided by d. That is, if n = dq + r for integers q and r with 0 < r < d, we write $r \equiv n \pmod d$ for the residue of n modulo d. Show that the residue modulo 7 of a (large) integer n can be found by separating the integer into 3-digit blocks $n = b(s)b(s-1)\dots b(1)$.(Note that b(s) may have 1, 2, or 3 digits, but every other block must have exactly three digits.) Then the residue modulo 7 of n is the same as the residue modulo 7 of $b(1) - b(2) + b(3) - b(4) + \dots \pm b(s)$. For example,
$n = 25,379,885,124,961,154,398,521,655 \pmod 7$
$\equiv 655 - 521 + 398 - 154 + 961 - 124 + 885 - 379 + 25 \pmod 7$ $\equiv 1746 \pmod 7$ $\equiv 746 - 1 \pmod 7$ $\equiv 745 \pmod 7 \equiv 3 \pmod 7$.
Explain why this works and show that the same trick works for residues modulo 13.

# Problem of the week, 137

A former colleague Bill Wardlaw (March 3, 1936-January 2, 2013) used to create a “Problem of the Week” for his US Naval Academy students, giving a prize of a cookie if they could solve it. One of them is given below.

Chain addition is a technique employed in cryptography for extending a short sequence of digits, called the seed to a longer sequence of pseudorandom digits. Quoting David Kahn (in Kahn on Codes, MacMillan, New York, 1983, p. 154), “the first two digits of the [seed] are added together modulo 10 [which means they are added and the carry is neglected] and the result placed at the end of the [sequence], then the second and third digits are added and the sum placed at the end, and so forth, using also the newly generated digits when the [seed] is exhausted, until the desired length is obtained”. Thus, the seed 3964 yields the sequence 3964250675632195… .

Periodic pattern

a. Show that this sequence eventually repeats itself.
b. Show that the sequence begins repeating itself with “3964”.
c. EXTRA CREDIT: How many digits are there before the first repetition of “3964”?

# Problem of the week, 148

A former colleague Bill Wardlaw (March 3, 1936-January 2, 2013) used to create a “Problem of the Week” for his US Naval Academy students, giving a prize of a cookie if they could solve it. One of them is given below.

Suppose p and q are each monic polynomials of degree 4 with real coefficients and the intersection of their graphs is {(1, 3), (5, 21)}. If p(3) – q(3) = 20, what is the area enclosed by their graphs?

# Problem of the week, 150

A former colleague Bill Wardlaw (March 3, 1936-January 2, 2013) used to create a “Problem of the Week” for his US Naval Academy students, giving a prize of a cookie if they could solve it. One of them is given below.

Let a, b, and c be real numbers and let f and g be real valued functions of a real variable such that $\lim_{x\to a} g(x) = b$ and $\lim_{x\to b} f(x) = c$.
a. Give an example in which $\lim_{x\to a} f(g(x)) \not= c$.
b. Give an additional condition on f alone and show that it
guarantees $\lim_{x\to a} f(g(x)) = c$.
c. Give an additional condition on g alone and show that it
guarantees $\lim_{x\to a} f(g(x)) = c$.

# mathematics problem 155

A colleague Bill Wardlaw (March 3, 1936-January 2, 2013) used to create a “Problem of the Week” for his students, giving a prize of a cookie if they could solve it. Here is one of them.

Mathematics Problem, #155

We can represent a triangle with sides of length a, b, c by the ordered triple (a, b, c). Changing the order of the sides doesn’t change the triangle, so (a, b, c), (b, a, c), (b, c, a), (c, b, a), (c, a, b), and (a, c, b) all represent the same triangle. To avoid confusion, let’s agree to write (a, b, c) with a < b < c. We say that a triangle <I (a, b, c) is integral if a, b, and c are integers. How many integral triangles are there with longest side less than or equal to 100 ?

# Mathematics Problem 154

A colleague Bill Wardlaw (March 3, 1936-January 2, 2013) used to create a “Problem of the Week” for his students, giving a prize of a cookie if they could solve it. Here is one of them.

Mathematics Problem, #154

Find the volume of the intersection of three cylinders, each of radius a, which are centered on the x-axis, the y-axis, and the z-axis. That is, find the volume of the three dimensional region

E = {(x,y,z) | x2 + y2 < a2, y2 + z2 < a2, z2 + x2 < a2}.

# Problem of the Week, #116

A colleague Bill Wardlaw (March 3, 1936-January 2, 2013) used to create a “Problem of the Week” for his students, giving a prize of a cookie if they could solve it. Here is one of them.

## PROBLEM 116

Bob invites Joe to see his new program, which prints $2 \times 2$ matrices with random integer entries. (These entries are as likely to be even as to be odd.) Joe, being a bit of a gambler, wants to bet Bob a dollar that the next matrix will have an even determinant. Should Bob take the bet? What is the probability that the next matrix will have an even determinant?

Note:
The determinant of the 2 x 2 matrix

$A = \left( \begin{array}{cc} a & b \\ c & d \\ \end{array} \right)$

is given by $det(A) = ad - bc$.

## ADVANCED PROBLEM 116A

What is the probability that the determinant of an $n \times n$ matrix with randomly chosen integer entries is divisible by the prime number $p$?
(Assume that the residues $0, 1, 2, ... , p-1$ are equally likely for each integer entry.)