I'm sure that the problem with the one that I don't understand is that I'm not interpreting the meaning of what is written correctly. The solution to the puzzle is given within the puzzle and Banneker also wrote a proportion at the end which yields the correct solution.

I've included below the puzzle - which is given in rhyme form (typical of the era), his solution, my interpretation of the meaning of key lines and my attempts at solutions - which aren't correct.

I would appreciate any help you could give.

Thanks,

John Mahoney, mahoneyj@sidwell.edu

Puzzle of the Dog and the Hare recorded by Benjamin Banneker in his Journal circa 1792

When fleecy skies have Cloth’d the ground
With a white mantle all around
Then with a grey hound Snowy fair
In milk white fields we Cours’d a Hare
Just in the midst of a Champaign
We set her up, away she ran,
The Hound I think was from her then
Just Thirty leaps or three times ten
Oh it was pleasant to see
How the Hare did run so timorously
But yet so very Swift that I
Did think she did not run but Fly
When the Dog was almost at her heels
She quickly turn’d, and down the fields
She ran again with full Career
And ‘gain she turn’d to the place she were
At every turn she gan’d of ground
As many yards as the greyhound
Could leap at thrice, and She did make,
Just Six, if I do not mistake
Four times She Leap’d for the dogs three
But two of the Dogs leaps did agree
With three of hers, nor pray declare
How many leaps he took to Catch the Hare.

(Answer)
Just Seventy two I did Suppose,
An Answer false from thence arose,
I doubled the Sum of Seventy two,
But still I found that would not do,
I mix’d the Numbers of them both,
Which Shew’d so plain that I’ll make Oath,
Eight hundred leaps the Dog did make,
And Sixty four, the Hare to take.

The work that follows is equivalent to the proportion 4/24 = 48/x solved to find x = 864.

Questions?

What is the mathematical meaning of the lines of the puzzle?
Why is 72 almost correct?
Why is 864 correct?
Where did the numbers 4 and 48 come from in the solution?

Statements
1) Initially the dog is 30 dog leaps from the hare.
2) The hare turns when the dog reaches the hare.
3) The hare gains 3 dog leap’s worth of yards at each turn.
4) The hare makes 6 turns.
5) 4 times the number of hare leaps = 3 times the number of dog leaps.
6) 2 times the length of each dog leap = 3 times the length of each hare leap.
7) The question: How many leaps does the dog make in order to catch the hare?

Statements 5) and 6) combine to state that the dog runs twice as fast as the hare.

Solution attempt #1:
Since initially the dog is 30 dog leaps from the hare and the dog runs twice as fast as the hare, then the dog reaches the hare in 60 dog leaps. At that instant the hare turns and is then three dog leaps ahead of the dog. The dog chases the hare and reaches the hare in 6 dog leaps. The hare turns again and, again, is three dog leaps ahead of the dog. This continues for a total of six turns until the dog finally catches the hare. Thus the dog has run 60 + 6*6 = 96 dog leaps. This answer is 1/9 of 864.
(perhaps this solution is flawed because it relies on fractional leaps? Check this.)

Solution attempt #2:
This attempt is based on interpreting the phrase “turn’d to the place she were” to mean the hare returned to the place where the hare started and at that point was an additional 3 dog leap’s worth of yards ahead of the dog.

Since initially the dog is 30 dog leaps from the hare and the dog runs twice as fast as the hare, then the dog reaches the hare in 60 dog leaps. At that instant the hare turns and is then 33 dog leaps ahead of the dog. The dog chases the hare and reaches the hare in 66 dog leaps. The hare turns again and, again, is 33 dog leaps ahead of the dog. This continues for a total of six turns until the dog finally catches the hare. Thus the dog has run 60 + 6*66 = 456 dog leaps.

A variation of this solution has the hare at each turn becoming increasingly further ahead of the dog by three dog leaps at each turn. So the solution becomes, in dog leaps: 60 + 66 + 72 + 78 + 84 + 90 + 96 = 546 dog leaps.

Note: Maybe the number of turns is wrong, but there is no arithmetic sequence with first term 60 and difference 6 whose sum is 864.

Well, I do not see how x = 864 solves 4/24 = 48/x. It does not. The solution is x = 48×6 = 288.

>1) Initially the dog is 30 dog leaps from the hare.

That's correct.

>2) The hare turns when the dog reaches the hare.

Not necessarily. All it'says is

>When the Dog was almost at her heels
>She quickly turn’d, and down the fields

The dog might have been close but not quite reaching the hare.

>3) The hare gains 3 dog leap’s worth of yards at each turn.

That's my understanding too.

>4) The hare makes 6 turns.

OK.

>5) 4 times the number of hare leaps = 3 times the number of
>dog leaps.

No, it's the other way around: 3 times the number of hare leaps equals 4 times the number of dogs leaps. The dog made fewer leaps. Every time it made 3 leaps, the hare made four leaps.

>6) 2 times the length of each dog leap = 3 times the length
>of each hare leap.

OK.

>7) The question: How many leaps does the dog make in order
>to catch the hare?

Right.

>Statements 5) and 6) combine to state that the dog runs
>twice as fast as the hare.

No, only 9/8 as fast.

I think the problem is much simpler than any of your solution would suggest. At the outset, the distance between the dog and the hare was 30 dog leaps. With 6 turns the hare added 18 leaps more. We may assume that the problem is equivalent to both running on a straight line with dog starting 48 leaps away.