## Friday, April 13, 2012

### Sport Puzzles

Sorry, no puzzle or newsletter last week. My days have been very busy with the Holidays, vacation, taxes and my kids' birthdays. For those who haven't read about my April family management miscalculations, here is my Due Date Math story.

This week I am coming back with a set of sport puzzles. You can answer one or all in a single comment below.

1. By non-obvious reasons I seem to be taking to the sports my kids learn. Perhaps I am envious and want to be more than a driver, perhaps it is watching them play that sparks the idea, perhaps it is just a convenience.  So, in parallel with my daughter, I started taking group tennis lessons. There are four of us in the group. A few weeks ago one of the players didn't come and we all found ourselves much more exhausted than usual. No wonder, we each had to work harder. By how much?
2. During the tennis lesson we spend 50% of the time learning to play and 50% gathering tennis balls all around the court. We are all beginners and balls are spread all around the court, frequently all around the town. Each of us picks a quarter of the court and gather all the balls onto his/her racket. Usually one holds the racket flat in one hand and picks the balls with another, carefully managing the balance of the racket as it is filled with balls. I have been thinking that a different approach is easier and faster. Place the racket on the floor in the center of the ball cluster and pick balls with both hands, filling the racket. Then move the racket to another cluster. What do you think?
3. My father is jogging every morning. He runs a few mins and then walks the same number of mins, then runs again etc. His running speed is twice than his walking speed. When he started his jog the other day he met an old friend running with a slower speed than my father. They waved. After completing one run and walk cycle my father met this friend again. What can be said about their relative speeds?
4. My friend's kids went to the sports camp during their school vacation. Camp instructions say that everyone should wear red on Monday, yellow on Tuesday, green on Wednesday, blue on Thursday and black on Friday. On one of the days 1/3 of the kids came in rainbow shirts as they were not able to find clothes of the required color in their closet. What day was it?

Your answers are accepted any time until midnight Eastern Time on Sunday, on our Family Puzzle Marathon.
Top image by Brooks Elliott, distributed under the CCL.

Tom said...

1. (Each had to work harder, to make up for the absence of one.) The pretty obvious answer would be that the remaining three had to do 33.3% of the work instead of 25% of the work, or, each had to do 33.3% (one-third) more work than before. But this does not address the nature and amount of the work being done, and that’s a big variable. It is very possible for the three players to choose to work only as hard as they usually did, or even less hard.

2. Depends very much on the true number of the tennis balls and the distribution of them. I think if there are many dozens of balls, randomly distributed, the second approach should be an improvement. Even more important is the path you follow between balls. In any case, running around chasing balls and picking them up is an important part of the exercise gained from playing such a game.

3. (It is unclear to me if they were running in the same direction or the opposite, but we’ll assume both are going counterclockwise as usual. We also know nothing about the size of the track, if it is a track or a set course. Perhaps Dad jogs “out and back?” Do they take the same course? Assume so.) Oh well. If Dad laps (meets) his friend in one run-walk cycle…I’m stuck. His friend could after be standing still, or moving very slowly, and I suspect the friend’s average speed is slower than Dad’s walking speed. What am I missing, guys?

4. Weekend? Don’t the clothes/uniforms get returned to their closets, or do they mysteriously disappear?

Strange questions and not really mathy, or I’m having a dumb day.

Anonymous said...

1. 33% more

2. I have a tennis racquet and only had four balls. I estimated that I could put about 16 balls on the racquet at once. I suppose it would be faster putting the racquet in one place and moving it around but I think part of tennis is the social interaction so that would be missed if people just picked up balls by themselves. If you really wanted to pick up balls faster and easier, I would suggest in getting pails and that way you would not have to balance the balls and could carry many more balls in a pail.

3. The friend is 1/4 slower than your dad.

4. I'm guessing Wed. since blue and yellow make green and they did not have the secondary color so they said blue and yellow make green.

Gurubandhu

Jerome said...

Problem One
=========
If each person makes the same complaint it means that everyone took turns being the only person on one side of the net. If the time for a full lesson is T minutes then each person will be a “single” for T/3 minutes. That only determines the fairness of the load.

Let us suppose that during the lesson, each person hits the ball n times. That means the single person will hit a total of 2*(n/3) balls. So the total number of hits is n/3 + n/3 + 2*n/3 = 4/3 n. The increase is n/3 hits. [(n/3)/n] * 100 = 33 1/3 %. Each person worked 1/3 more than if there were 4 people.

Problem Two
==========
My first inclination is to be a jerk and say it would be much better to collect the balls in one or two of those reusable shopping bags. I’m just not sure how to collect the balls. Your second method might be superior in the case where the balls seem to be clustered close together and near the center.

It might be very inefficient if the balls are near the net or the back of the court on a diagonal of the corner of the court furthest from the center or around the perimeter of your quarter. It all depends on what the clusters look like.

Problem 3
=======
First we have to settle direction and the shape of the jogging route that both men follow. Since they wave at each other,

I make the first assumption that they are running in opposite directions. I make this assumption because each of them sees the wave made by the other.

The next assumption I make is the path they follow is circular.

The third is that your father must cover at least a little more than half the circular path. You did say that his friend’s speed was less than your father’s. Now to the relative speeds.

Suppose the ½ way point is x. Your father goes a distance less than x (call it a) when he meets his friend. The speed at which your father travels is (x + a) / ( 3T ) . His friend is travelling at a speed of (x – a) / (3 T) where T = your father’s walking time.

The ratio of their speeds (x + a) / ( x – a).

Problem 4
========
I’m a little stuck on this one. It may be that everyone owns only a T shirt (white) or a rainbow colored one. I presume you mean that 1/3 of the people do not have a black shirt and a rainbow colored shirt will take the place of black, in which case it’s Friday. Black absorbs all the rainbow colors. It would be better if they had to wear white on Friday.

Jerome's wife said...

Answering #2 & #4. On #2 I would choose to place the racket flat on the ground near each clump of balls, using both hands to pile them in the center of the racket. My reasons for choosing this choice was because it is the surest way to not lose the balls and waste time retrieving them.

On #4 the Sports camp kids must wear one specific color from Mon thru Fri, and 1/3 end up wearing rainbow shirts one of those days, my guess would be that the day was Friday. I'm not sure why it should be Friday,but my thought is that black is not considered a color, which has nothing to do with the fact it was chosen as a color to be worn on Friday. So unless it is tricky in the sense that black is not a color...I'm really not sure, but took a stab at a guess.

Annie said...

1. Not sure how the game was set up with only 3 (took turns playing singles?, 2 on 1.? I'll just say everyone had to give 8.3% more. Usually each player carries 25% and with only three they now have to carry 33.3% though not sure it's that simple.

2. Aren't there buckets to put the balls in? That would be my first choice. If not, I definitely like your approach. No balancing required, two hand retrieval, and faster movements. Or you could work in twos with one person holding the racket and another putting the balls on it. This way you wouldn't have to keep running back to the racket as your partner could hold the racket steady and quickly follow you around. Granted you'd have a larger area to cover but I think it's more efficient.

3. Your father's friend's speed is 1 1/2 times your father's walking speed.

x = walking speed of Dad 2x = jogging speed of Dad y = friend's
speed

xt + 2xt = y(2t)
3x=2y
y=3/2x

4. Friday 1/3 of the children wore rainbow shirts. Black is the absorption of the entire spectrum of light so black actually is the entire rainbow.

Jerome said...

I've spent most of today trying to figure out if I should say anymore. I decided to do it because one should really.

Problem 2 is really trouble. It may be more practical to do the balls one at a time, but maybe not. The difficulty is that you all are beginners. I think you might find it hard to balance the ball on the racket. A pro doesn't think about it. He/she has handled a racket so often that they can balance any kind of situation on their rackets. That's what's wrong with using both hands and running back and forth after picking up two (or maybe 4 balls). It could much more efficient but the pyramid could be awfully diffiult to carry without spilling the whole bunch which means you'll have to start all over again.

Problem 4
I would really like to say that it is Saturday or Sunday because the rainbow does not really create black at all.

If I must choose a week day then of course it is black. But I'm troubled no matter what the answer and I'm sure I could get this one wrong if there's a trick.

TracyZ said...

puzzle 1) when one of the four tennis players doesn't come, 3 players are doing the work that 4 players usually do. Each of these three players is working 4/3 as much as usual, or 33.3% more than when four players are there.
--------
puzzle 2) For collecting up all the tennis balls, Maria, I like your approach of placing the racket on the ground and then gathering the balls with two hands. Your approach means that you may be traveling a greater distance on the court collecting balls, but you will likely be bending down to pick up balls fewer times than with the initial method, both because you can now pickup balls with both hands, and because to me, it seems hard to keep balls on the racket while bending down to pick up balls and standing back up. With the original method, I think you might end up having to pickup some balls multiple times in one round because they fall off your racket. This is less likely to happen with your new approach.
-------
puzzle 3) To solve this problem, I think that one needs to assume that your father and his friend are jogging the same route, such as on a track and making at least one or more loops on this route. This isn't explicitly part of the problem but I am not sure how to solve it otherwise.

He runs x minutes at a speed of 2y miles/hour
He then walks x minutes at a speed of y miles/hour
Distance = velocity * time
His distance = his running distance + his walking distance
His distance = 2y miles/60 minutes * x minutes + y miles/60 minutes * x minutes
His distance (d1) = 2xy/60 + xy/60 = 3xy/60 = xy/20

His friend's exercise:
He runs 2x minutes at a speed of z miles/hour
His distance (d2) = 2xz/60 = xz/30

For them to meet:
d1 = d2
xy/20 = xz/30
y/20 =z/30
y = 2/3z
----------------
puzzle 4) The camp kids didn't know what to color wear on Friday (the day for wearing black) because black is the absence of color, not a color itself.

Maria said...

I know it is a vacation week for many and it is impressive to see so many answers.

Here are the answers I had in mind:
1)With 4 people in the group we each work 1/4=25% of the time. With 3 people we each work 1/3=33.3%. To answer what proportion of the time we work more we have to divide (33.3-25) by 25 = 8.3/25 = 1/3 or 33.3%

Tom, Gurubandhu, Jerome, Annie and TracyZ answered this one correctly and will get a puzzle point each.

2)This puzzle was just intended to originate a discussion and point our attention to the unexpected math of every little thing we do. Who knew that math can surface up during a tennis lesson. All the discussion participants get a puzzle point: Tom, Gurubandhu, Jerome, Jerome's wife, Annie and TracyZ.

3)This is the most complex math puzzle from the bunch. My mistake - I didn't specify that they both run in the same direction. But many figured this out.

My intended solution was:
my dad has the speed of x when running and x/2 when walking
he runs time t with x, then walks time t with x/2
his distance is tx + tx/2 = 3/2 tx
his time is 2t
in the same time his friend walks the same distance with the speed of y
2ty

so 2ty = 3/2 tx
y = 3/4 x
friend's speed is 3/4 my dad's running speed or 3/2 my dad's walking speed.

Only Annie agreed with me on this solution and receives a puzzle point.

If we assume that they run in the opposite directions around on a track, then
instead of writing 2ty = 3/2 tx
we need to write
2ty = S - 3/2 tx where S is the total length of the track

4)Here you all took me much more seriously than intended. This is a real story and I was just musing about the least common color of kids' clothes. Turns out it is green.
We usually buy red, yellow, white, black, purple or pink for the girls. And blue, black, gray, white for the boys. Green is less common.
I should disclose that the story took place outside of my home state of MA where green Celtics shirts are very popular eve for little kids :)
Gurubandhu got this one correctly and get another puzzle point for it.

Have a great tax-vacation-spring week.

TracyZ said...

Hi Maria,
I have been looking at yours and Annie's solutions to problem 3, and I believe that mine is essentially the same, just with slightly different notation:
you wrote that:
your dad's friend had speed y while walking (I think you have meant running as was stated in the problem)
the ratio of the running speeds: y = 3x/4

Annie wrote:
dad's friend speed y while running
dad's speed 2x while running
ratio of the running speeds: y=3x/2

I wrote:
dad's friend speed z while running
dad's speed 2y while running
ratio of the running speeds: y = 2z/3
... or as I am re-writing it now: z = 3y/2

while I think is the same as Annie's and your equations, just with a different assignment for y.

Am I totally off here? :) (I sometimes am.... )

Thanks,
TracyZ

Maria said...

Hi TracyZ,

No, I was totally off here.
Your solution is absolutely correct. Somehow I missed it. Definitely another well-deserved puzzle point.

And, by the way, thank you for your recommendation of the math exhibit at Smithsonian in DC. Anyone reading this who finds themselves in the DC, check out mathalive.com

TracyZ said...

Maria,
I was at the MathAlive exhibit yesterday -- it was great! It has lots of interactive activities for kids (designing skateboards, "riding" snowboards, reducing commuter traffic on a bridge, saving a city after a power outage, operating a mars rover, and more) and they can learn some math along the way too. Mine spent three hours there and would have stayed much longer.

Jerome said...

I hate to be a grouch here. I seldom argue with an answer, but I think problem 3 has another twist to it.

If you start by assuming that the path is circular, you really cannot tell anything if they are travelling in opposite directions (which I think is a perfectly natural assumption). The 3/2 that you get I think comes from the linear assumption (ie they are travelling in the same direction). It just bothers me. Am I wrong?

Maria said...

Here is what I think happens when two people run in the opposite direction:

my dad has the speed of x when running and x/2 when walking
he runs time t with x, then walks time t with x/2
his distance is tx + tx/2 = 3/2 tx
his time is 2t
in the same time his friend walks with the speed of y
his distance is = 2ty

Assuming that S is the total length of the track, we have
my dad's distance = S - other man's distance

3/2 tx = S - 2ty

we can't really solve this as we have too many unknowns :)

Anonymous said...

Maria, I am not happy with your answer to the first question. When 4 people are playing tennis each player theoretically hits 1/4 of the balls. If only 3 people are playing, then each player hits 1/3 of the balls. So the difference between is 1/3 less 1/4 ie 1/12. So I believe that the answer is 1/12 or .08%

Maria said...

Dear Anonymous -
you are absolutely right - it is 1/12 or 0.08 or 8% of the total time of the lesson.

However if we take it relative to the usual part each of the students participates, that is 1/4 or 0.25 or 25%, the we will get

(1/12) / (1/4) = 1/3

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