## A Barnum Puzzle

Posted by Karl Sharman on January 27, 2011 – 11:05 am

This one is via PT Barnum, probably by Lloyd.

A cat and dog run a race, 100 feet straight away and back again. The dog leaps three feet at each bound and the cat but two, but then she makes three leaps to his two. Now, under those circumstances, what are the probabilities or possibilities in favour of the one that gets back first?

Oh, and if you didn’t read the first line- it comes via Barnum – A sucker born every minute – that Barnum…..

January 27th, 2011 at 11:18 am

based on initial calculations, they both cross at the same time

January 27th, 2011 at 11:28 am

Unfortunately, I fear the cat will win. The dog’s pace has him wasting time at the 100 feet divide line with a partial bound, whereas the cat can gracefully make both 100ft distances without wasting any distance on an extra bound (100ft/3 gives a remainder, while 100ft/2 does not).

January 27th, 2011 at 11:41 am

I think BearSprite is right.

This makes the following assumption: the dog and cat take the exact same time to stop and turn around.

The dog will land at the 99ft mark and still have to take another jump which puts him at 102 for the turn around.

The cat will land on the 100ft mark and immediate turn around. So the dog is now 2 feet behind. And will still be 2 feet behind when the cat gets back to the starting point.

I would think though, since the dog has more mass, it would take hime longer to slowdown at the turnaround point, putting him even further behind.

Poor dog!

January 27th, 2011 at 11:48 am

Hmmm!

The question leads us to assume that the cat is female and the dog is male but that isn’t actually stated.

Are you leading us up the garden path Mr S?

January 27th, 2011 at 12:00 pm

You can even make a further assumption that the dog does a shorter 1ft leap at the 99ft mark so he doesn’t go over 100ft, and he’ll still come second, regardless of time taken to turn around, or time taken to cover any excess distance.

January 27th, 2011 at 12:01 pm

Regarding my comment #5 – that may not hold true if the bound takes less time. All in all, however, the cat’s got it in the bag.

January 27th, 2011 at 4:19 pm

Man’s best friend has better motivation, the cat will take his sweet time getting started and comming back, assumning you can convince the cat to run in a straight line and back just to beat the dog. The cat has better odds but my money is on the dog.

January 27th, 2011 at 6:55 pm

It mainly depends on which aninmal runs the fastest.

By considering parabolic trajectories and assuming that each animal leaps up with the same initial angle, then the leap length is proportional to the average speed. So the dog runs 1.5 times as fast as the cat.

The dog should win easily. The extra few feet is unimportant.

January 27th, 2011 at 8:05 pm

It depends on when the cat and dog switch the number of feet per leap. If the dog takes 2 leaps at 3 feet each and then 47 leaps at 2 feet and the cat does takes 2 leaps at 2 feet each and then 32 leaps at 3 feet each, the dog would have a total of 49 leaps and the cat 34. The cat would win in this case assuming the timing per leap is the same for the cat and the dog.

But, if the dog takes 32 leaps at 3 feet and then 2 at 2 feet and the cat takes 47 at 2 feet and 2 at 3 feet, the dog would win.

It ultimately depends on when they change the number of feet per leap and if they change at the same time and if it takes the same amount of time per foot or per leap.

January 28th, 2011 at 9:17 am

Chris, I have taken time out of my busy schedule to scald a cat and a dog’s tails at the same time, to ensure speedy running, and in a consistent direction (away from the tail), and using high speed cameras check the parabolic trajectories of the animals boundings…… The RSPCA have impounded my equipment and I haven’t yet seen the photographic proof so I cannot verify your information….yet.

Lane – what? The animals don’t change their running styles – it’s consistent throughout the course. Unless one of the animals is secretly not what we think it is…? Ask Dual Aspect, he might know something.

January 28th, 2011 at 4:10 pm

Hey Karl, are you hinting that I might be on the right track?

I’m assuming that there’s some kind of trick in the question because you have hinted that we might get suckered into a wrong answer.

The only thing I can see that could be a mis-direction is the uncertainty over whether the cat takes three leaps to the dog’s two, or the dog takes three leaps to the cat’s two because we don’t know which is male and which is female.

If the dog is female then it will definitely cross the line first.

If the cat is female then as stated by others it’s a close run thing and partly comes down to whether the dog is able to make a partial leap to complete the 100 metres, and if so whether that partial leap of 1 foot takes 1/3 of the time of a three foot leap.

January 28th, 2011 at 4:54 pm

Dual Aspect – got it – I’ll post the full answer after the weekend!

January 29th, 2011 at 5:22 am

One of my posts has vaporised. I had corrected the parabolic trajectory stuff. It should have concluded that the dog runs sqrt(3/2) ≈ 1.225 times as fast as the cat.

However, now that I’ve actually read the question, I see that the cat and dog don’t leap up at the same angle and that both animals run at the same speed.

The dog has to overrun the 100′ mark by 2′ on way out, and so has 102′ to run on the way back. So the cat wins by 4′.

January 31st, 2011 at 7:12 am

The correct answer as printed by Barnum / Lloyd:-

Now, the cat wins, of course. It has to make precisely 100 leaps to complete the distance and return. The dog, on the contrary, is compelled to go 102 feet and back. Its thirty-third leap takes it to the 99-foot mark and so another leap, carrying it two feet beyond the mark becomes necessary. In all, the dog must make 68 leaps to go the distance. But it jumps only two thirds as quickly as the cat, so that while the cat is making 100 leaps the dog cannot makes quite 67.

But the puzzle turns upon the possibilities of the race. Just let us suppose that the cat is a tom-cat, and the dog is a…. female … so the dog really goes 9 feet to the cats 4. As the dog finishes the race in 68 leaps, the cat will have gone but 90 feet and 8 inches.

Well spotted Dual Aspect!

February 1st, 2011 at 2:59 pm

I would have said the cat would win because it was probably being chased by the dog and would have ended up as a snack if it didn’t .