Teacher resources and professional development across the curriculum

Teacher professional development and classroom resources across the curriculum

Mathematics: What's the Big Idea?

# Math Puzzlers

### Workshop #1 Puzzler — Banana Split

A vendor sells 1/2 of his stock of bananas plus 1/2 banana.

He then sells 1/2 of his remaining stock plus 1/2 a banana.

Again he sells 1/2 of his remaining stock plus 1/2 a banana. This depletes his stock of bananas? How many did he have to start with?

### Workshop #2 Puzzler — Planet of the Apes

Three monkeys walk into a motel on the Planet of the Apes and ask for a room. The desk clerk says a room costs 30 bananas, so each monkey pays 10 bananas towards the cost.

Later, the clerk realizes he made a mistake, that the room should have been 25 bananas. He calls the bellboy over and asks him to refund the other 5 bananas to the 3 monkeys. The bellboy, not wanting to make a mess dividing the 5 bananas three ways, decides to lie about the price, refunding each monkey 1 banana, keeping the other 2 bananas for himself. Ultimately each monkey paid 9 bananas towards the room and the bellboy got 2 bananas, for a total of 29 bananas. But the original charge was 30 bananas.

Where did the extra 1 banana go?

### Workshop #3 — Corey Camel's Bananas

Consider the case of Corey Camel - an enterprising, albeit eccentric owner of a small banana plantation in a remote desert oasis.

Corey's harvet, worth it's weight in gold, consists of 3000 bananas. The market place where the stash can be cashed in is 1000 miles away. However, Corey must walk to the market, and can only carry up to 1000 bananas at a time. Furthermore, being a camel, Corey eats one banana during each and every mile she walks (so Corey can never walk anywhere without bananas). How many bananas can Corey get to the market?

### Workshop #4 — Banana Farmer's Dilemma

A banana farmer has six monkeys who hate each other.They must be kept in pens to separate them from the bananas as well as from each other.

The farmer has created six pens of equal size using 13 equal lengths of fencing. The pens are organized in a row.

Early one morning the farmer discovers someone has stolen one of the lengths of fencing.

How can the farmer reorganize the fencing to make six new pens of equal size with the 12 remaining lengths of fencing?

### Workshop #5 — Grudy's Bunch

• When it's your turn, you split any one bunch into two smaller bunches. (One banana is still a bunch. No fractional bananas.)
• You may not split a bunch into two equal bunches.
• If you can't move, you lose!

Decide: Would you rather go first or second?

Extensions: What if you started with a different number of bananas?

Example

Bob and Elise begin with a bunch of 8 bananas.

Bob splits 8 into 5 and 3 (he's not allowed to do 4 and 4)

Elise may split either the 5 or the three. She chooses the five and splits it into 2 and 3.

Now Bob sees two threes and a two. He can't split the two (it would make two equal bunches), but he may split either of the threes.

Elise has no choice. She can't split the twos. She can't split the one. So she splits the remaining three.

Bob can't move. He loses. Elise wins.

### Workshop #6 — Heavy Banana

You have 12 bananas.

11 are exactly equal in weight.1 is heavier than the others.You want to find the heavy banana.

Using a balance scale, what is the smallest number of weighings you needto find the heavy banana?

### Workshop #7 — I'll Take Bananas for \$100...

You're back on the Plant of the Apes, and you're a contestant in a game show.

There are three doors, and behind one door is an enormous pile of bananas. Behind the other two, there are just banana peels. (Remember: Bananas are gold on the Planet of the Apes!)

The host asks you to pick one. You do, but before he opens the door, he opens one of the other two doors that you did not chose. He shows you a room full of peels — no bananas.

He offers you a choice:

Stick with your original choice OR switch to the remaining unopened door.

Which should you do?

### Workshop #2

This answer is courtesy of Hilda Douglas.

Regarding the monkeys and the bananas, it is a question of possesion vs. payment. The motel owner possess 25 bananas. Each of the three monkeys possesses 1 banana totalling 3 bananas. The bellboy possesses two bananas making a grand total of thirty banas.

### Workshop #3

Thanks to Gail Lauinger for this week's banana puzzler!

### And now, the best solution we've been able to find:

If Corey just picks up a load of 1,000 bananas and heads out across the desert, she will eat them all up by the time she gets to the other side. She will also leave 2,000 bananas, unused, to rot back at the oasis. The trick is to use those 2,000 bananas as fuel to get the remaining 1,000 bananas as far across the desert as possible, before Corey makes her final dash for the market.

Corey needs to eat five bananas per mile so long as she is trying to ferry more than 2,000 bananas. Later, when she's hauling between 1,000 and 2,000 bananas, she needs three bananas per mile. And after that, she only eats one banana per mile.

To understand why, let's start at the beginning.

Corey is standing there in the oasis with 3,000 bananas. She picks up the first 1,000. Say she carries them just one mile into the sand, eating one banana. She could drop 999 bananas there, but then she couldn't walk back. So, being a camel with foresight, she drops 998 bananas and keeps one to eat on the return trip.

Now she can pick up the second 1,000 bananas and do the same thing, dropping 998 at the one-mile marker and shambling back to the oasis.

With the third load, there's no return trip: all her bananas have been moved one mile.

How many did she eat up? Five: two on the first round trip, two on the second, and one on the last trip, which is one-way.

She could keep this up, one mile at a time, for 200 miles, by which time she would have used up 1,000 bananas. Or she could just take the first load 200 miles, drop 600 bananas, go back, pick up the next 1,000, etc. Either way, she will find her self at the 200-mile marker with 2,000 bananas.

(Note that there are no monkeys or hungry humans out there in the sand dunes, and no other camels, either. Corey feels her bananas will be safe when she drops a load in the desert and goes back for more.)

Once she has the 2,000 bananas out in the desert, Corey the Mathematical Camel reasons that she now needs three bananas per mile to push her stash farther: 1 round trip for the first load of bananas and 1 one-way trip for the second load. Either with her calculator or with mental math, she determines that she will use up the second 1,000 bananas moving the supply forward 333 1/3 miles. She can either proceed in one-mile increments, or go the whole 333 1/3 miles at once, or anything in between. In the end, Corey finds herself with 1,000 bananas 533 1/3 miles (200 + 333 1/3) into her journey.

It's hot, but Corey takes a deep breath, picks up the 1,000 bananas, and slogs on. This time she can just keep going with no return trips, because she hasn't left any bananas in the desert - just in her stomach.

433 2/3 miles farther on, and lighter by 433 2/3 bananas (she's a nibbler), Corey pads out of the desert and into the market, where a mob of camel-lovers and mathematicians is waiting to pay her handsomely for the 533 1/3 bananas (1,000 - 433 2/3) she has left. She even sells that last 1/3 of a banana to a souvenir hunter from the Annenberg Channel.

### Workshop #4

A hexagon configuration provides six new pens of equal size forthe angry monkeys.

### Workshop #5

This solution is from TimErickson, content guide for the fifth workshop and sixth workshops.

You could solve the problem simply by playing. You'll discover,for example, that "ones" and "twos" are effectively out of the game.You'll also soon see that if it's your turn and you have a bunch offour (and nothing else) you're doomed.

You can use that information and your growing experience to findout that if you go first, you can always win if you first split thebunch of eight into seven and one. Then, whatever your opponent does,you can leave him or her with a bunch of four. Then you're home free.The only thing you can do with four is three-and-one (since two-twois illegal -- equal bunches are forbidden) and then it's your turn.You split the three into two and one, and your opponent cannot move.

There is no obvious pattern. That's OK. The key thing for you --and for students -- is to think about all of the possibilities andfigure out a way to keep track of them.

### If you'd like a more complete solution, read on.

In the diagram, each circle represents a position in the game. Thenumbers represent the bunches of bananas. For simplicity, ones andtwos (which are indivisible) don't appear here. Thus, 3-1-1-3 issimply 3-3 -- two bunches of three bananas each.

The arrows represent moves. So this diagram represents everythingthat could possibly happen in the game!

You can work backwards to figure out what's best. We know that fora "3", the next player wins. That means that a 3-3, for example,favors the previous player (you don't want it to be your turn whenyou see it) because it can only lead to a "3." 4-3 is good, then,because if it's your turn, you can leave your opponent in that awful3-3 position (or simply "4").

You can analyze all of the positions that way. The red circles aregreat if it's your move (you're the next player). The green circlesare great if it's the other player's move (you're the previousplayer).

So, since 8 is red -- next player wins -- you want to be the nextplayer. So go first. But you'd better move to "7" (making a 7-1split) or your opponent will have one of those excellent redpositions (6 or 5-3) and will stick you with a loss (by making 4 or3-3, respectively).

You can analyze just about any nim game this way. Nim games arestrategy games in which generally take things away (BalloonRide in the Family Math book is an example). This game(Grundy's) is unusual in that you're not taking away bananas, justthe chance to split the bunches.

Students often don't realize that there is no probability in thesegames. They're completely deterministic. If you know the strategy andget your choice of whether to go first or second, you will win everytime. Thus nim games are great for developing strategy in teams: thegoal is not to win the game but to develop and describe an unbeatablesystem.

### Workshop #6

1. Split the bananas in half - 6 in each bunch. Weigh each bunch. The heavier bunch contains the heavy banana.

2. Split the heavier bunch in half - 3 in each bunch. Weigh each bunch. The heavier bunch contains the heavy banana.

3. Split the remaining 3 bananas apart. Weigh two of the bananas. If they weigh the same, the heavier banana is the one you did not weight. If one weighs more than the other, then that is the heavy banana.

Thanks to Hilda Douglas for another correct answer!

Mathematics: What's the Big Idea?