The Garden Grows...

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Two Elevators

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The Locker Problem

a classic

When students returned to school after winter break, they decided to see what would happen if they went into school one at a time.  As each student went into the school they would open or close the lockers according to a set of directions.  The first student

Answer:  This is a classic problem and it you do a smaller problem you find that the lockers that are square numbers are open (ie. 1,4,9,16,25, 36, 49, 64, ...).  One of my students gave a great explanation... here it is:

  If a locker is changed an even amount of times then it will go back to being closed.  If it is changed an odd amount of times, then it will be open.  What kind of number has an even number of factors (how many factors a locker number has determines how many times it is changed)?  Most numbers have an even amout of factors, because for every factor there is a matching one that when multiplied will equal the locker number.  Except for square numbers because for one of their factor pairs the numbers are the same, and so those two factors, being exact same, are only counted as one.  That would make the amount of factors for those numbers  be odd.   

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The Skateboard Run

Some of our students are setting up a skateboarding course for a competition.  The course is 80 yards long.   They are using cones to have to skate around as a slalom run. If the cones are placed 8 yards apart and starting at the start line and the last is placed at the finish line how many are needed for the entire course?

Answer:  There are 11 cones needed.  One every 10 yards and then one to begin the race with.

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A Whole Bunch of Problems!!!

1. For the year 1994, no date exists for which the product of the month and day equals the number named by the last two digits of the year.  Last year, 3-31-93 met this criterion.  During this century (1900 -1999), which year would have the most possibilities?(2pts)

2. While driving the mathematics team to a contest, the driver noticed that the bus odometer read 21,912 miles.  Being a lover of mathematical curiosities, she noticed that this number was a palindrome, reading the same backward and forward.  The team sponsor said, "I bet it will be a long time before that happens again!"  However, just 105 minutes later, the bus driver reported that the mileage again was a palindrome!  How fast was the bus traveling? (3 pts)

3. A certain kind of folder was priced at 50¢ but found no buyers.  The manager of the store decided to reduce the price.   Within a few days, she had sold her entire stock of these folders for a total of $31.93.  What was the reduced price of the folder? (2pts)

Answers:   1) 1924.  It has the most factors.  2) Approximately 63 mph.  the next palindrome is 22022 or 110 miles.  So in 105 minutes or 1.75 hours, they drove 110 miles. Just divide to find the answer.  3) $0.31 or $0.01.   The number must be a factor of 3193.

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2 Flat Tire Problems

  Mrs. Adams woke up one morning to find that one the wheels of her car had a completely flat tire.  Despite this she set off in her car and drove 100 miles to visit a another school.  She then drove 100 miles home.   She did not repair or inflate the flat tire.  How did she manage to make the journey?

   Four high school students arrived late for school, explaining to Mrs. V, the principal, that their car had suffered a flat tire on the way there.  How did the clever principal immediately discern that the students were not telling the truth?

Answer:

    #1 The flat tire was the spare... so it was not needed to drive at all.  I told you to think laterally.

     #2 The clever principal took each student to a different room and asked each one which tire was flat.  Of course each said a different tire and they were caught.

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Plates, bottles and Glasses

Here's a classic from the book Cyclopedia of Puzzles, written by Sam Loyd in 1914. 

If a bottle and a glass balance with a pitcher, a bottle balances with a glass and a plate, and two pitchers balance with three plates, how many glasses will balance with a bottle?

Answer: It will take 5 glasses.

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Wendy's Family?

Wendy's family has fewer than ten children.  She has the same number of sisters as she has brothers, but her brother Andy hastwice as many sisters as he has brothers.  How many boys and how many girls are in the family

Be Good to Your Family

Answer: There are four girls and three boys.   Because Wendy has the same number of sisters as brothers, we need to test only the possibilities in which Wendy has the same number of sisters and brothers up to four each. If Wendy has five sisters and five brothers, then the family has more than ten children.   Once we know Wendy's siblings, we know that Andy has one more sister than Wendy and one fewer brother.  We would then need to check to see whether the number of Andy's sisters is twice that of brothers.

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Gillian's Construction Co.

Gillian's Construction Company is building a drainage ditch that is 800 feet long, 6 feet wide, and 5 1/2 feet deep.  If a truck can carry 2000 cubic feet of dirt, how many truckloads will be needed to carry away the dirt?

Be Good to Your Family

Answer: 14 Truck loads.  The volume of dirt taken away is 800 x 6 x 5.5, or 26,400 cubic feet.  This amoount divided by 2000 gives 13.2 truckloads of dirt.

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Mr. Titen and His Long Drive

Mr. Titen was traveling with his parents.  He noticed a sign that read, "Highland Park 111 miles;   Bud Lake 173 miles."  He continued his book, figuring that he had a while before his parents stopped for lunch in Bud Lake.  Sometime later, his mother said, "Look, Bud Lake is exactly twice as far away as Highland Park."  How many miles had they driven while Mr. Titen was reading?

Be Good to Eachother

Answer: 49 Miles.  Use a drawing.  If  Bud Lake is 173 miles away and Highland Park is 111 miles away, then the two towns are 62 miles apart.   To be twice as far from Bud Lake as Highland Park means that my family must be 62 miles away from Highland Park.  The distance from the first sign to Highland Park was 111 miles, and 111 - 62 = 49. 

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We Can Make Any Amount of Change

Here's a tough one:

What is the least number of coins that you need, to pay exactly, for any amount up to but not including $1.00?

Be Good to Your Teachers

Answer: Ten coins (not using 50¢ pieces), 4 pennies, two nickels, 1 dime, 3 quarters; nine if a fifty-cent piece is used to replace two of the quarters.

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The Vacation Package

#2) Jona has only 20 minutes before he must go to bed, but he wants to play a soccer game that he found on the Internet.  The computer downloads at a rate of 58,000 bytes an hour. The game that Max wants to play consists of 8,000 bytes.  How much playing time to the nearest second does Max have after the game downloads before he has to be in bed?(2pt)

#3) Fluffy was born on 24 June 1983. On her first birthday.   Fluffy's age in cat terms was equivalent to that of a 15-year-old human.  When she turned 2, her age in human years was 24.  At the end of her third year, her human age was 28, and when she was 4, her human age was 32.  She continued to age 4 years with each birthday. What year will Fluffy turn 100 in cat years?(2pt)

#4) Because Ms. Drozd wanted to win the Berryville bicycle race, she trained hard for a week, including the weekend.  Each day, she rode 3 miles farther than she ahd the day before.  If she rode a total of 126 miles, how far did she ride on the last day before the race?(2pt)

#5) Mrs. Scott bought 100 pounds of potatoes that were 99 percent water.  After the potatoes were left outdoors for a day, they were 98 percent water.   What was the weight of the slightly dihydrated potatoes? (3pt)

Be Good to Your Teachers

Answer: 1) 853.  The total number is 1634; let x be the number of boys, and let x+72 be the number of girls.  Then x + x + 72 = 1634; therefore, x = 781 and 781 + 72 is the number of girls.

2) Jona has 11 minutes and 43 seconds to play.   Dividing 58,000 by 60 minutes gives about 966.7 bytes per minute.  Dividing 8,000 by 966.7 gives 8.28 from 20 minutes leaves 11.72 minutes wich is 11 minutes and 43 seconds.

3) On June 24, 2003, Fluffy can blow out 100 candles if she can stay awake long enough.

4) 27 miles on the last day. One strategy to use is guess and check.  An algebraic approach would let x be the number of miles biked on the first day.  Then the equation would be x + (x +3) + (x + 6) + (x+9) + ( x + 12) + (x + 15) + (x + 18) = 126

5) 50 pounds. If the potatoes were originally 99% water, that means they contained 1% solids, or 1 pound of solids.  The amount of solids stays the same no matter how much water is lost.  On the second day, if the potatoes were 98% water, 2% of their wiehgt is in solids.  Because the potatoes contain one pound of solids, if w = the weight of the potatoes at eh end of the second day, then 0.02 x w = 1.  Solving for w, we get w = 50

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Can You Crack the Code?

Here is an unusual safe.  Each of the buttons must be pressed only once in the correct order to open it.  (Every button must be pressd.) The last button is marked F.   The number of moves and the direction is marked on each button.  Thus 1U would mean one move up, whilst 1L would mean one move to the left.  Which button must be pressed first? Give its exact location.  Hint: it is somewhere in the first row? Give the order of the buttons. (2pt)

Be Good to Your Teachers

Answer: The first Button is 2D in the third column.  From there you'll have to figure out the rest... sorry.

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What's the pattern??

Fill up this square with the numbers 1 to 5 so that no row, column or diagonal line of five squares uses the same number more than once.  What number should replace the question mark?

Be Good to Your Teachers

Answer: 5 should replace the question mark.

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