Reasoning
Stage 5's clue tells us that 150m was left. So the distance left at the start of Stage 4 must have been:
Dist4 = Dist4 + 75 + 150
—————
2
Which simplifies to give Dist4 = 450m.
Stage 3's clue tells us that 450m was one quarter of the remaining distance, so Dist3 = 1,800m.
Stage 2's clue tells us that the distance at the start of Stage 2 was:
Dist2 = Dist2 + 35 + 1800
—————
2
Which simplifies to give Dist2 = 3,670m.
Stage 1's clue tells us that 3,670m was half the overall distance, which means the entire ride was 7,340m.
Double-Checking
Starting with 7,340m:
Stage 1: cycled 3,670m, leaving 3,670m
Stage 2: cycled 1,870m, leaving 1,800m
Stage 3: cycled 1,350m, leaving 450m
Stage 4: cycled 300m, leaving 150m
Stage 5: cycled 150m, leaving 0m
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Puzzle 6
Consider an arrow in flight towards a target.
At any given moment of time, a snapshot could be taken of this arrow. In this snapshot, the arrow would not be moving. Let us now take another snapshot, leaving a very small gap of time between them. Again, the arrow is stationary. We can keep taking snapshots for each moment of time, each of which shows the arrow to be stationary. Therefore the overall effect is that the arrow never moves, however it still hits the target!
This is a classic paradox, attributed to Zeno of Elea, a Greek philosopher from Italy. Great minds over the centuries have pondered this paradox, and the scope of a solution is beyond the space available here. It is not even clear that a solution to the paradox actually exists.
??
Puzzle 7
What is the minimum number of queens required on a chessboard such that all squares are attacked?
Mamma: "Let me think, Tommy. Well, our three ages add up to exactly seventy years."
Tommy: "That's a lot, isn't it? And how old are you, papa?"
Papa: "Just six times as old as you, my son."
Tommy: "Shall I ever be half as old as you, papa?"
Papa: "Yes, Tommy; and when that happens our three ages will add up to exactly twice as much as to-day."
Tommy: "And supposing I was born before you, papa; and supposing mamma had forgot all about it, and hadn't been at home when I came; and supposing--"
Mamma: "Supposing, Tommy, we talk about bed. Come along, darling. You'll have a headache."
Now, if Tommy had been some years older he might have calculated the exact ages of his parents from the information they had given him. Can you find out the exact age of mamma?
Mamma's Age, Amusements In Mathematics, Henry Ernest Dudeney.
Hint
The answer isn't a whole number of years, and algebra might be required.
Answer
29 years 2 months.
Reasoning #1
This answer is taken directly from the original book.
The age of Mamma must have been 29 years 2 months; that of Papa, 35 years; and that of the child, Tommy, 5 years 10 months. Added together, these make seventy years. The father is six times the age of the son, and, after 23 years 4 months have elapsed, their united ages will amount to 140 years, and Tommy will be just half the age of his father.
Reasoning #2
Here's my answer, with a little algebra.
If we call Tommy T, Mamma M and Papa P we can see that:
"our three ages add up to exactly seventy years" gives us:
(1) T + M + P = 70
"Just six times as old as you" gives us:
(2) P = 6 x T
In an unknown number of years (Y) "Shall I ever be half as old as you" gives us:
(3) P + Y = 2 x (T + Y)
and "our three ages will add up to exactly twice as much as today" gives us:
(T + Y) + (M + Y) + (P + Y) = 140
which can be written as
(4) T + M + P + 3Y = 140
We can see from (4) and (1) that
3Y = 70
so
(5) Y = 70 ÷ 3
Using (2) and (5) in (3) we have
P + Y = 2 x (T + Y)
6 x T + 70 ÷ 3 = 2 x (T + 70 ÷ 3)
4 x T = 70 ÷ 3
(6) T = 70 ÷ 12
We can now use (6) in (2) to see that:
P = 6 x T
P = 6 x 70 ÷ 12
P = 70 ÷ 2
And using the values for T and P in (1) we have:
T + M + P = 70
70 ÷ 12 + M + 70 ÷ 2 = 70
Multiply throughout by 12 to give:
70 + 12 x M + 420 = 840
12 x M = 840 – 420 – 70
12 x M = 350
M = 350 ÷ 12
So:
Tommy = 70 ÷ 12 = 5.83333 = 5 years 10 months. Papa = 70 ÷ 2 = 35 = 35 years. Mamma = 350 ÷ 12 = 29.1666 = 29 years 2 months.
