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(reminder: all quotes here are fiddled, probably.)


反擊/反抗, 非私了



天地間真滋味, 唯靜者能嘗得出;
天地間真機括, 唯靜者能看得透;
天地間真情景, 唯靜者能題得破.
作熱鬧人, 說孟浪話, 豈無一得, 皆偶合也.

The frog problem

standupmaths, Can you solve The Frog Problem? (youtube)


1. When the frog jumps it randomly chooses from one of the possible landing places.
2. The frog keeps jumping forwards until it crosses the river.
3. If the maximum is n jumps and the minimum is 1 jump to cross the river, what is the expected number of jumps?

The answer should be an explicit formula, not a non-recusive relation.
Someone said in the comments that the answer is the harmonic series.

Let E(n) be the expected number of jumps in where the frog has n possible spaces to jump to at the beginning. (i.e. n-1 lotus leaves and 1 spot for the opposite bank).
E(n) = 1/n + 1/(n-1) + 1/(n-2) + ... + 1

I cannot quite follow how he gets E(n) = 1/n * (1 + E(n-1)) + (n-1)/n * E(n-1), though I get the same answer E(n) = 1/n + E(n-1).

Here is my line of thought:

E1 = 1
E2 = 1/2*1 + 1/2*(1 + E1)
E3 = 1/3*1 + 1/3*(1 + E1) + 1/3*(1 + E2)
E(n) = 1/n*1 + 1/n*(1 + E1) + 1/n*(1 + E2) + ... + 1/n*(1 + E(n-1))

Grouping the terms we get
E(n) = 1 + 1/n*(E1 + E2 + ... E(n-2) + E(n-1))

Now let's look at E(n-1). Following the similar argument we have
E(n-1) = 1 + 1/(n-1)*(E1 + E2 + ... + E(n-2)), so
E1 + ... + E(n-2) = (n-1)*E(n-1) - (n-1)

Putting into E(n) we get E(n) = 1/n + E(n-1).

Penelope to Ulysses

From Ovid's Heroides, translated by Harold Isbell (1990):

Penelope to the tardy Ulysses:
do not answer these lines, but come, for
Troy is dead and the daughters of Greece rejoice.
But all of Troy and Priam himself
are not worth the price I've paid for victory.
How often I have wished that Paris
had drowned before he reached our welcoming shores.
If he had died I would not have been
compelled now to sleep in my cold bed
complaining always of the tiresome
prospect of endless nights and days spent working
like a poor woman at my tedious loom.


You are my refuge and my home, my husband.
Have a father's concern for your son;
let us see a son's concern for your father
who waits now for you to close his eyes.
Just remember, I was a young girl when you left;
if you came at once you would find an old woman.

* * *

Clare Pollard performs from her new translation of Ovid's Heroides
Phaedra to Hippolytus: (youtube)