Three Teasers of Different Sorts
Problem 1: In the center of a circle 9 ft. in radius is a frog. It begins to jump in a straight line to the circumference of the circle. its first jump is 4 ft. 6 in., its second 2 ft. 3 in., and it continues to jump half the length of the preceding jump. How many jumps does the frog make to get out of the circle?
Problem 2: My friend Skwareham, who sold carpets, said that if he had to multiply two numbers together he usually squared their average instead. The answer was always too big but was near enough, he said. I told him that if he was all that keen on squaring instead of multiplying, he might as well do a little bit more and get the answer right. what else ought he to do?
Problem 3: The cost of hiring the bus was shared equally among all those who went on the outing. It was a 34-seater and the bill came to 8₤ 11s. 7d. How many empty seats were there?
T.R. (Bellingham) offered these comments on these problems: "Your current first and third problems of the week remind me of how backward we Americans are, even compared to the British, who are stereotypically thought to be old-fashioned and resistant to change. The British adopted decimal currency a long time ago (around 40 years ago I think) so your third problem would be as incomprehensible there as here. But they also went metric for measurements a long time ago--it is only Americans who won't let go of feet and inches. Also: the answer to the frog problem would seem to depend on some details not included in the problem--does "get out of the circle" mean that all parts of the frog have to be outside the circle? In that case we need to know the length of the frog from front to back and the point on the frog from which the measurements are being made. Depending on those details the answer changes quite a lot."
Source: R. Wesley's (ed.) Mathematics For All, 1956, pp. 62, 244, 435
Hint: For the first, draw a picture and model the jumps...and what do you notice?
For the second, try to express the situation algebraically...and what can you include to adjust his squaring formula?
For the third, you need to do some converting...how many pence in a shilling and how many shillings in a pound...then, think...
Solution Commentary: As a final hint on the first, if it remains unsolved...do some reading on Zeno's motion paradoxes.
As to the second, consider the algebraic expression [(a+b)/2]2 - [(a-b)/2]2...
For the third, even the book gave a wrong answer or an incomplete answer. As the total cost was 2059 pence and 2059 = 29 x 71, the book concluded that 29 people went on the outing with 5 seats empty, each person paying 71 pence (or 6s. 11p.). BUT...the book's author missed the possibility of 2059 = 1 x 2059...or 1 person went on an expensive outing with 33 seats empty!