dan's math@home - problem of the week - archives
 
 
Problem Archives page 17
Problems Only. For answers & winners click here.
 
1-10 . 11-20 . 21-30 . 31-40 . 41-50 . 51-60 . 61-70 . 71-80 . 81-90 . 91-100
101-110 . 111-120 . 121-130 . 131-140 . 141-150 . 151-160 . 161-170 . 171-180
181-190 . 191-200 . 201-210 . 211-220 . 221-230 . 231+ . prob index
 
161- Dimples All Over
162 Your Fav Subject?
163 - Riding The Train
164 - Three More Sqrs
165 -- Here's To 2003!
166 -- Happy And Sad
167 ExerciseEfficiently
168 Arranged by Hight
169 - Pay For Grades?
170 - Area's The Same
 
 
Problem #161 - Posted Friday, December 27, 2002
Dimples All Over ! (back to top)
A regulation golf ball is spherical and has 384 dimples, arranged in a triangular pattern.
Most of the dimples are surrounded by six other dimples, but some are surrounded by
only five. How many dimples have only five neighbors? Give a mathematical proof if possible.
Please explain your steps and reasoning.
 
 
Problem #162 - Posted Monday, January 6, 2003
Your Favorite Subject ? (back to top)
Last year, near the end of my "Survey of Math" course, we had time for one more topic:
Polyhedra, Divisors, or Curves. I decided to let my fifteen students vote their preferences :
Each student submitted their 1st, 2nd, and 3rd choice topics. There were 3 ways to count
the ballots, defined as follows: (1) Plurality: The topic with the most first-choice votes ("prefs") wins.
(2) Instant Runoff: Eliminate the third-place topic (in number of prefs), and add the 2nd-choice votes of
that topic to prefs of the first two. (3) Point system: Two pts for each 1st choice, one for each 2nd choice.
If I counted the votes by plurality, Polyhedra won, but by the instant runoff method,
Divisors was the choice. And under the point system, Curves was the winner!
How was this possible?! Give a scenario with 15 actual preferences.
 
 
Problem #163 - Posted Saturday, January 25, 2003
Riding The Train (back to top)
Two students were on a commuter train from New York to Boston. One said, "The trains
coming from Boston pass us every five minutes. I wonder, how many Boston trains arrive
in New York per hour, given equal speeds in both directions?" "Twelve, I figure," said the
second, "sixty minutes divided by five." The first disagreed. Who was right?
 
 
Problem #164 - Posted Thursday, February 6, 2003
Three More Squares . . . (back to top)
"I found three perfect squares in arithmetic progression!" said Fred. "That's the smallest
solution. But I have three that are even," Steven countered, "and also a formula for an
infinite family of progressions! For each triple, I see the first two are the squares of
p^2 - q^2 - 2pq and p^2 + q^2." a) What were Fred's small and Steven's even solutions?
b) What's the square root of that third square in each p, q progression?
 
 
Problem #165 - Posted Sunday, February 16, 2003
Here's To a peaceful 2003 ! (back to top)
a) In the sequence : 12345678910111213 . . . what position does 2003 first appear?
b) How many different ways can 2003 be written as the sum of three squares?
c) Find the smallest number that's the sum of three squares in this many ways.
Rearrangement does not count as different.
 
Problem #166 - Posted Friday, February 28, 2003
: ) Happy And Sad : ( . . . (back to top)
Each day, Stevie is either happy or sad.
If he is happy one day, then four times out of five he is happy the next day.
If he is sad one day, then he is sad the next day one time out of three.
In the long run, what are the chances that Stevie is happy on any given day?
 
 
Problem #167 - Posted Sunday, March 16, 2003
Exercising Efficiently. .. (back to top)
Excess Exercise Ernie can do 20 pullups, or 30 situps, or 40 pushups, in one minute.
Doing one pullup burns 1.5 calories; one situp burns 1.8 cal; a pushup burns 1.2 cal.
Ernie is too 'pressed' for time; you need to design him two Efficient Exercise Efforts:
(i) Panic Workout: Lasts seven minutes, burns exactly 300 calories, with a total of 200 reps,
(ii) Full Session: At least 30 min, burning at least 1200 calories, with as few reps as possible.
Each workout must contain at least one rep(etition) of each exercise.
State how many reps of each type, and total reps, for each workout.
 
 
Problem #168 - Posted Monday, March 31, 2003
Arranged by Hight ?. . . (back to top)
No, that's not a misspelled word. The hight h(p) of a polynomial p(x) is defined as the sum of
the degree n and the absolute values of the coefficients: h(x^2 - 3x + 5) = 2+1+3+5 = 11.
a) If n is allowed to be 0, how many polys p(x) are there with integer coeffs and hight
not exceeding 1, 2, 3, 4, 5 ? (That's five answers. For the first 3 give an actual list of the p(x).)
b) What's the hight of a monic cubic poly with roots a, b, and 3 ? (Ans. in terms of a and b.)
"Monic polynomial" means leading coeff. is 1. Explain steps and reasoning.
 
 
Problem #169 - Posted Friday, April 11, 2003
Pay For Grades?!. . (back to top)
Sarah's parents decide to reward her (with money!) for her school grades. Sarah has five
classes (English, History, Japanese, Math, and Science). She will get $15.00 for each A,
$10.50 for a B, $7.50 for a C, $1.50 for a D. Of course, she gets nothing for an F.
The last three semesters Sarah has avoided F's, and received the same amount of money
(a whole number of dollars) each semester, although the grade distribution of A's, B's, C's, D's
was different each time. . a) How much money did Sarah receive after each semester?
b) What were her grades and semester G.P.A.* each time? (Don't worry about which grades
she got in which classes.) * GPA is avg grade pts per class: 4 pts for A, 3 for B, 2 for C, 1 for D, 0 for F.
(I'm not condoning or condemning this pay idea.)
 
 
Problem #170 - Posted Monday, April 21, 2003
Area's The Same ! . (back to top)
Two things come to mind when looking at right triangles:
The Pythagorean Theorem relating the three sides, a, b, c ;
and the formula for the area A. Sometimes, all four of
these quantities are integers: a 3-4-5 triangle has area 6.
What is the smallest natural number A that's the area of two
different integral right triangles? (Switching a and b isn't different.)
What's the smallest natural number A that's the area of three i.r.t.'s?
What's the smallest A (if there is one) that's the area of four i.r.t.'s?
Show your steps, reasoning, or search method. Give sides and areas.
Sum of squares of legs
= square of hypotenuse
 
 
 
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Problem Archives Index
 
Probs & answers . 1-10 . 11-20 . 21-30 . 31-40 . 41-50 . 51-60 . 61-70 . 71-80 . 81-90 . 91-100
Problems only . . . 1-10 . 11-20 . 21-30 . 31-40 . 41-50 . 51-60 . 61-70 . 71-80 . 81-90 . 91-100
Probs & answers . 101-110 . 111-120 . 121-130 . 131-140 . 141-150 . 151-160 . 161-170 . 171-180
Problems only . . . 101-110 . 111-120 . 121-130 . 131-140 . 141-150 . 151-160 . 161-170 . 171-180
Probs & answers . 181-190 . 191-200 . 201-210 . 211-220 . 221-230 . 231+
Problems only . . . . 181-190 . 191-200 . 201-210 . 211-220 . 221-230 . 231+
 
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