Lesson Plan Week 11

Homework Problem

Answer: Impossible

Question 1

You have 8 marbles with one heavier than the other 7 even though it looks the same. You have a weighing scales. Using only 2 weighings how can you find which marble is the odd one?

Solution: Split into 3,3 and 2 and weigh them.

Question 2: Sirepinski Carpet

So, we’re going to start off with a square – draw a nice big square on graph paper. For reasons that will become clear in a while, make it 27 boxes wide and 27 boxes high.

Now follow these simple steps:

1. Break your square up into 9 equal squares, as on the right, and colour in the middle square.

Question: How much of the square have you removed? So how much of the square is left?

2. Now take each of the 8 smaller squares that are left, divide each of these into 9 equal squares, and colour in

the middle square in each.

Question: How much of each square have you removed? So how much of Carpet 1 have you removed? And how much is left?

Further Question: How much of the original square is now left? Can you work out your answer without counting little squares? If you can, check your answer using the little squares.

3. Now take each of the 64 small squares, divide each of

theseinto 9 equal squares, and colour in the middle square in each.

Question: How much of each square have you removed? So how much of Carpet 2 have you removed? And how much is left?

Further Question: How much of the original square is now left?

The BIG Question:

How much of the original square would be left if you did this process:

  • One more time?
  • Ten more times?
  • A hundred more times?
  • Forever?

Question 3: Dominoes

Dominoes are 2x1 rectangles with 0 to 6 spots on each square. All 28 possible pairs of numbers of spots (including doubles) are represented. The game is played by forming a chain in which squares of adjacent dominoes have equal numbers of spots.

Challenge 1.

All the dominoes in a set are laid out in a chain, so that the number of spots on the ends of adjacent dominoes match.

If one end of the chain is a 5, what is at the other end?

Challenge 2.

In a set of dominoes, all those in which one of the squares has no spots are discarded. Can the remaining dominoes be arranged in a chain?

Question 3

Can you draw the shape (attached separately) without taking your pen off the paper and without tracing any line more than once?

Answer by first asking can about the other shapes in attachment.

Question 4: Probability

  • Suppose I flip two coins without letting you see the outcome, and I tell you that at least one of the coins came up heads. What is the probability that the other coin is also heads?

Solution1/3.
For two coins, there are four possible outcomes: HH, HT, TH, and TT. Since we know that at least one was a head, we can eliminate TT. Of the remaining three possibilities, only 1 allows the second head: HH.

  • In a box you have 13 white marbles and 15 black marbles. You also have 28 black marbles outside the box. Remove two marbles, randomly, from the box. If they are of different colors, put the white one back in the box. If they are the same color, take them out and put a black marble back in the box. Continue this until only one marble remains in the box. What color is the last marble?

SolutionThe last marble will be WHITE.
Since marbles can only be taken out in pairs and you started off with an odd number of whites there is always going to be one white left over that you'll keep putting back in the box until it's left on its own.

  • Henry has been caught stealing cattle, and is brought into town for justice. The judge is his ex-wife Gretchen, who wants to show him some sympathy, but the law clearly calls for two shots to be taken at Henry from close range. To make things a little better for Henry, Gretchen tells him she will place two bullets into a six-chambered revolver in successive order. She will spin the chamber, close it, and take one shot. If Henry is still alive, she will then either take another shot, or spin the chamber again before shooting.
    Henry is a bit incredulous that his own ex-wife would carry out the punishment, and a bit sad that she was always such a rule follower. He steels himself as Gretchen loads the chambers, spins the revolver, and pulls the trigger. Whew! It was blank. Then Gretchen asks, "Do you want me to pull the trigger again, or should I spin the chamber a second time before pulling the trigger?"
    What should Henry choose?

SolutionHenry should have Gretchen pull the trigger again without spinning.
We know that the first chamber Gretchen fired was one of the four empty chambers. Since the bullets were placed in consecutive order, one of the empty chambers is followed by a bullet, and the other three empty chambers are followed by another empty chamber. So if Henry has Gretchen pull the trigger again, the probability that a bullet will be fired is 1/4.
If Gretchen spins the chamber again, the probability that she shoots Henry would be 2/6, or 1/3, since there are two possible bullets that would be in firing position out of the six possible chambers that would be in position.

  • You are out one night having a few milkshakes when a shadowy figure approaches you. "Wanna play a game?" he inquires. "I'll toss two coins and look at them. If they are both tails then I'll show you both and we'll flip again. On the other hand, if they are not both tails, this is how we'll play. I'll slide one coin across to you. It has to be a head or a tail right? I'll give you $3 if it's a head and you give me $2 if it's a tail. Wanna play?"
    Would you play with this stranger and if you decide yes, who will win or is it a fair game?

SolutionIf you play you will lose. Outcomes are TT (no play) HH HT or TH. In the last 3 cases the stranger will keep a head, sliding you a tail on two occasions and a head on one occasion. He will win twice as often as you. He will win $4 to your $3 so in the long run you will lose.

Question 5

A clown has 3 balls. He weighs 14 stone and the balls weigh 1 stone each. He needs to cross a brige that can only take 16 stone. He has 20 mins to get across and it takes him 10 mins to walk across. How does he do it??

Hint: Clown

Solution: Juggles