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# CaptainEd

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1. ## Jelly beans join the clean plate club

Here’s a tiny observation about what the next to last step looks like.
2. ## Waiting, again II

much more clearly stated than my babbling. I’m tickled that I’ve shown that it can be evaluated one flip at a time, based merely on the parity of contiguous Hs. here is my argument, expressed by plagiarizing your expression: Let e be the expected number of flips from the initial (even) state There are two states that are easy to analyze and cover all the possibilities: E is the initial state, it also represents the state of having seen an even number of H (including zero), since the beginning or the most recent T. O is the odd state, representing the fact of having seen an odd contiguous run of H. State E requires e more flips. In this state, H changes to state O, while T remains in state E State O requires o more flips. In this state, H changes to state E, while T terminates with a win. That allows us to write an expression for x as the sum of these terms, weighted by their respective probabilities, all 1/2. e = 1/2{1+e} + 1/2{1+o} o = 1/2{1+e} + 1/2 substitute o into e e = 1/2{1+e} + 1/2{1+1/2{1+e} + 1/2} = 1/2{1+e} + 1/2+1/4+e/4 + 1/4 = 3/2 + 3e/4 e/4 = 3/2 e = 6

6. ## Waiting, again

Gardner sets high standard in many ways. I was a child reading Childrens Activities and a few years later I was enjoying hexaflexagons and later mathematical games. I was kneeling behind you in worship. i enjoy the puzzles here, and sometimes I don’t understand something that is obvious to anyone else. I think I may have a touch of ambiguity flu. Keep on puzzling, Bonanova!
7. ## Waiting, again

Thank you Bonanova, sorry to be so dense
8. ## Waiting, again

I want to be sure I’ve got this right. There are six (presumably distinguishable) dice. I want to demonstrate that each is capable of showing all six faces. Paul lets me roll all six (and tabulate which individual dice showed which numbers) charges \$1 for the combined roll, and pays \$50 when the tabulations show each die has shown all six faces. Peter has me roll one die at a time, charges \$1 for the individual roll, tabulates the results, and pays \$20 when each die has shown all six faces. OR... my Termination condition is seeing all six numbers on the table at once. Paul has me roll all 6 dice each time, and only pays me if I roll a full straight (123456). Peter lets me roll one die at a time, once I’ve rolled all six dice, he lets me improve my hand by rolling a single die that duplicates another one, and pays once all six numbers are showing.

Perhaps
10. ## Adapting a classic crossing puzzle

Good job slashpuzzler! And neat puzzle BMAD!
11. ## Two can tango

Nice job, araver! Nice puzzle, bonanova
12. ## Soldiers in a field

Maybe this is clearer and more accurate than my previous try. Point one:
13. ## Line splitting

It's been a long time. Here's a recharacterization that recognizes a much smaller search space: Overview Definitions and additional constraint LineSplittingMValues.xlsx BonanovaSequence.xlsx
14. ## Count the Flags

Gavin, I agree, a permeable chain is necessary. Here’s a simplified (and still incomplete) proposal: