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 $\lim_{x\to a} f(g(x)) = c$.

Problem of the Week, #119

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 119

A spider puts on $8$ identical socks and $8$ identical shoes (and of course the spider has $8$ feet). In how many different ways can the spider do this, given that on each foot, the sock has to go on before the shoe?

Problem of the Week, #118

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 118

Suppose that you draw numbers $x_1,x_2,\dots$ from $[0,1]$ randomly until $x_1+x_2+\dots +x_n$ first exceeds $1$. What is the probability that this happens on the fourth draw? That is, what is the probability that $x_1+x_2+x_3 < 1$ and $x_1+x_2+x_3+x_4 > 1$?

PROBLEM 118A

In the above problem, what is the expected number of draws until the sum first exceeds $1$?

Problem of the Week, #117

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 117

What is the largest number of regions $r(n)$ that a plane is divided into by $n$ straight lines in the plane?
Give $r(n)$ as a function of $n$ and explain why your answer is correct.

PROBLEM 117A

What is the largest number of regions $r(n, d)$ that $d$-dimensional Euclidean space is divided into by $n$ hyperplanes?
Give $r(n, d)$ as a function of $n$ and $d$, and find formulas for $b(n, d)$, the number of regions that are bounded, and for $u(n, d)$, the number of regions that are unbounded.
Of course, $r(n, d) = b(n, d) + u(n, d).$ Explain why these numbers are correct.