The Buried Treasure
(
THE PROFESSOR'S PUZZLES)
The problem of the Buried Treasure was of quite a different character. A young fellow named Dawkins, just home from Australia, was introduced to the club by one of the members, in order that he might relate an extraordinary stroke of luck that he had experienced "down under," as the circumstances involved the solution of a poser that could not fail to interest all lovers of puzzle problems. After the club dinner, Dawkins was asked to tell his story, which he did, to the following effect:—
"I have told you, gentlemen, that I was very much down on my luck. I had gone out to Australia to try to retrieve my fortunes, but had met with no success, and the future was looking very dark. I was, in fact, beginning to feel desperate. One hot summer day I happened to be seated in a Melbourne wineshop, when two fellows entered, and engaged in conversation. They thought I was asleep, but I assure you I was very wide awake.
"'If only I could find the right field,' said one man, 'the treasure would be mine; and as the original owner left no heir, I have as much right to it as anybody else.'
"'How would you proceed?' asked the other.
"'Well, it is like this: The document that fell into my hands states clearly that the field is square, and that the treasure is buried in it at a point exactly two furlongs from one corner, three furlongs from the next corner, and four furlongs from the next corner to that. You see, the worst of it is that nearly all the fields in the district are square; and I doubt whether there are two of exactly the same size. If only I knew the size of the field I could soon discover it, and, by taking these simple measurements, quickly secure the treasure.'
"'But you would not know which corner to start from, nor which direction to go to the next corner.'
"'My dear chap, that only means eight spots at the most to dig over; and as the paper says that the treasure is three feet deep, you bet that wouldn't take me long.'
"Now, gentlemen," continued Dawkins, "I happen to be a bit of a mathematician; and hearing the conversation, I saw at once that for a spot to be exactly two, three, and four furlongs from successive corners of a square, the square must be of a particular area. You can't get such measurements to meet at one point in any square you choose. They can only happen in a field of one size, and that is just what these men never suspected. I will leave you the puzzle of working out just what that area is.
"Well, when I found the size of the field, I was not long in discovering the field itself, for the man had let out the district in the conversation. And I did not need to make the eight digs, for, as luck would have it, the third spot I tried was the right one. The treasure was a substantial sum, for it has brought me home and enabled me to start in a business that already shows signs of being a particularly lucrative one. I often smile when I think of that poor fellow going about for the rest of his life saying: 'If only I knew the size of the field!' while he has placed the treasure safe in my own possession. I tried to find the man, to make him some compensation anonymously, but without success. Perhaps he stood in little need of the money, while it has saved me from ruin."
Could the reader have discovered the required area of the field from those details overheard in the wineshop? It is an elegant little puzzle, and furnishes another example of the practical utility, on unexpected occasions, of a knowledge of the art of problem-solving.
Answer:
The field must have contained between 179 and 180 acres—to be more exact, 179.37254 acres. Had the measurements been 3, 2, and 4 furlongs respectively from successive corners, then the field would have been 209.70537 acres in area.
One method of solving this problem is as follows. Find the area of triangle APB in terms of x, the side of the square. Double the result=xy. Divide by x and then square, and we have the value of y2 in terms of x. Similarly find value of z2 in terms of x; then solve the equation y2+z2=32, which will come out in the form x4-20x2=-37. Therefore x2=10+(sqrt{63})=17.937254 square furlongs, very nearly, and as there are ten acres in one square furlong, this equals 179.37254 acres. If we take the negative root of the equation, we get the area of the field as 20.62746 acres, in which case the treasure would have been buried outside the field, as in Diagram 2. But this solution is excluded by the condition that the treasure was buried in the field. The words were, "The document ... states clearly that the field is square, and that the treasure is buried in it."
Informational
