CHAPTER 6 Flashcards
MARRIED COPULES MAY CROSS A RIVER
Find the smallest number (x) of persons a boat may carry so that (n) married couples may cross a river in such a way that no woman ever remains in the company of any man unless her husband is present. Also find the least number of passages (y) needed from one bank to the other. Assume that the boat can be rowed by one person only
- n x y table?
- formula?
- n = 2, x = 2, y = 5
- n = 3, x = 2, y = 11
- n = 4, x = 3, y = 9
- n = 5, x = 3, y = 11
- n>5, x = 4 and y = 2n -1
14 - 7 - 7 scoring
A, B, and C participate in a track meet, consisting of at least three events. A certain number of points are given for first place, a smaller number for second place, and a still smaller number for third place. A won the meet with a total score of 14 points; B and C are tied for second with 7 points apiece. B won first place in the high jump. Who won the pole vault assuming no ties occurred in any event?
- who won pole vault?
- total score?
- scoring?
- A B C table?
Total score of 3 players = 28
events:
1st event high jump
2nd event pole vault
and 2 others event
score:
4 (first place), 2 (second place), 1(third place)
score per event:
A = 2 + 4 + 4 + 4 =14
B = 4 + 1 + 1 + 1 = 7
C = 1 + 2 + 2 + 2 = 7
SIMPLEST SOLUTION IN INTEGERS
Find the simplest solution in integers for the equation:
1/x^2 + 1/y^2 = 1 / z^2
- values of x y z?
- formula of x and y?
x = 15, y = 20, z = 12
x= m^4 - n^4
y = 2mn (m^2 + n^2)
y = 2mn (m^2 - n^2)
CENSUS TAKER, AGE AND STREET NUMBER
Maynard the Census Taker visited a house and was told, “Three people live there. The product of their ages is 1296, and the sum of their ages is our house number.” After an hour of cogitation, Maynard returned for more information. The house owner said, “I forgot to tell you that my son and grandson live here with me.” How old were the occupants and what was their street number?
- How old were the occupants
- what was their street number?
ages = 1, 18, and 72
street number = 91
PRODUCT OF 4 INTEGERS
Prove that the product of 4 positive integers cannot be a perfect square.
proof
2 positive squares cannot differ by 1
BYZANTINE BASKETBALL
In Byzantine basketball there are 35 scores which are impossible for a team to total, one of them being 58. Naturally a free throw is worth fewer points than a field goal. What is the point value of each?
free throw?
field goal?
free throw = 8
field goal = 11
ADD OR MULTIPLY INTEGERS, ANSWER WAS THE SAME
Gherkin Gesundheit, a brilliant graduate mathematics student, was working on an assignment but, being a bit absent-minded, he forgot whether he was to add or to multiply the three different integers on his paper “He decided to do it both ways and, much to his surprise, the answer was the same. What were the three different integers?
1, 2, 3
ADAM, BROWN, CLARK SAME NUMBER OF ACRES
Three farmers, Adams, Brown and Clark all have farms containing the same number of acres. Adams farm is most nearly square, the length being only 8 miles longer than the width. Clark has the most oblong farm, the length being 34 miles longer than the width. Brown’s farm is intermediate between these two, the length being 28 miles longer than the width. If all the dimensions are in exact miles, what is the size of each farm?
Adam?
Brown?
Clark?
Adam = 40x48,
Brown = 32x60
Clark = 30x64
ICE SCREAM SALE 1960, 1961, 1962
1960 and 1961 were bad years for ice cream sales but 1962 was very good. An accountant was looking at the tonnage sold in each year and noticed that the digital sum of the tonnage sold in 1962 was three times as much as the digital sum of the tonnage sold in 1961. Moreover, if the amount sold in 1960 (346 tons), was added to the 1961 tonnage, this total was less than the total tonnage sold in 1962 by the digital sum of the tonnage sold in that same year. Just how many more tons of ice cream were sold in 1962 than in the previous year?
361 tons
THREE RECTANGLES WITH SAME AREA
Three rectangles of integer sides have identical areas. The first rectangle is 278 feet longer than wide. The second rectangle is 96 feet longer than wide. The third rectangle is 542 feet longer than wide. Find the area and dimensions of the rectangles.
area?
Area = 1,466,640 sq. ft.
Rectangular dimensions:
1080 by 1358
1164 by 1260
970 by 1512
WILLIE SOLD ALL HIS LEMONADE, DIME, NICKEL QUARTER
When little Willie had sold all his lemonade he found he had $7.95 in nickels, dimes and quarters. There were 47 coins altogether and, having just started to study geometry, he noticed that the numbers of coins satisfied a triangle inequality, i.e., the sum of any two denominations was greater than the third. How many of each were there?
dime?
quarter?
nickel?
Dime = 20
Quarter = 23
Nickel = 4
PIGGY BANK, HALF DOLLAR, DIME PENNIES
There are 100 coins in a piggy bank totaling $5.00 in value, the coins consisting of pennies, dimes and half dollars. How many of each are there?
half dollar = 1
dimes = 39
pennies = 60
ENGINEER CONSULTANT
Every year an engineering consultant pays a bonus of $300 to his most induslrious assistant, and $75 each to the rest of his staff. After how many years would his outlay be exactly $6,000 if all but two of his staff had merited the $300 bonus, but none of them more than twice?
no. of years?
no. of staff?
no. of years = 8
no. of staff = 7
DECIMAL POINT TO DOT MULTIPLY
In European countries the decimal point is often written a little above the line. An American, seeing a number written this way, with one digit on each side of the decimal point, assumed the numbers were to be multiplied. He obtained a two-digit number as a result, but was 14.6 off. What was the original number?
5.4
5 ⋅ 4 = 20
3 DIGIT
A certain 3-digit number in base 10 with no repeated digits can be expressed in base R by reversing the digits. Find the smallest value of R.
- 3-digit number in base 10?
- 3-digit number in base R?
- R?
R = 14
834 base (10) = 438 base (14)
