Question

For many years now Baron Münchhausen has gone to a lake every day to hunt ducks. Starting on August 1, 2000, he says to his cook: "Today I shot more ducks than two days ago, but fewer than a week ago." For how many days can the baron say this? (Remember, the baron never lies.)

Answer 1

lets call the number of ducks shot on the i'th day \(d_i\), then we get:\[
\begin{align}
\text{day 1:}\quad&d_1\\
\text{day 2:}\quad&d_2\\
\text{day 3:}\quad&d_3\\
\text{day 4:}\quad&d_4\\
\text{day 5:}\quad&d_5\\
\text{day 6:}\quad&d_6\\
\text{day 7:}\quad&d_7\\
\text{1 Aug 2000 day 8:}\quad&d_6<d_8<d_1\\
\text{day 9:}\quad&d_7<d_9<d_2\\
\text{day 10:}\quad&d_8<d_{10}<d_3\\
\text{day 11:}\quad&d_9<d_{11}<d_4\\
\text{day 12:}\quad&d_{10}<d_{12}<d_5\\
\text{day 13:}\quad&d_{11}<d_{13}<d_6\\
\text{day 14:}\quad&d_{12}<d_{14}<d_7\\
\text{day 15:}\quad&d_{13}<d_{15}<d_8\\
\text{day 16:}\quad&d_{14}<d_{16}<d_9\\
\end{align}\]days 15 and 13 cannot both be true, so he can only say the statement up to day 14, which means he can say it for 7 days.

Answer 2

august1, 2000: x
day 2: y
day 3: z
day 4: f
day 5: t
day 6: u

\[x < y < z < f < t < u\]

backward
day -1: y - c
day -2: x - h; u + p
day -3: t + k
day -4: f + e
day -5: z + d
day -6: y +b
day -7:x + a

x - h < u

But u must be greater than x. so it's only 7 days

Answer 3

Nice try guys, but...
The correct answer is six days.
I'm looking for the flaws in your logic...

Answer 4

It all depends on how you interpret:
"For how many days can the baron say this?"

Does this include 1 August 2000 or not?
i.e. is it "for how many days" or is it "for how many MORE days"?