
Here is a very well written Open Source Discrete Math book:
Discrete Mathematics  An Open Introduction[^]
You can download the PDF for free.
Interactive Book(Read it online)
Even if you're not interested in the book, check out how nicely the interactive book is done.
Discrete Mathematics  interactive book[^]
About The Book: The text began as a set of lecture notes for the discrete mathematics course at the University of Northern Colorado. This course serves both as an introduction to topics in discrete math and as the "introduction to proofs" course for math majors. The course is usually taught with a large amount of student inquiry, and this text is written to help facilitate this.
Four main topics are covered: counting, sequences, logic, and graph theory. Along the way, proofs are introduced, including proofs by contradiction, proofs by induction, and combinatorial proofs. An introductory chapter covering mathematical statements, sets, and functions helps students gain familiarity with the language of mathematics, and two additional topics (generating functions and number theory) are also included.
Here's a nice puzzle example (you've probably seen before)
Quote: While walking through a fictional forest, you encounter three trolls guarding a bridge. Each is either a knight, who always tells the truth, or a knave, who always lies. The trolls will not let you pass until you correctly identify each as either a knight or a knave. Each troll makes a single statement:
Troll 1: If I am a knave, then there are exactly two knights here.
Troll 2: Troll 1 is lying.
Troll 3: Either we are all knaves or at least one of us is a knight.
Which troll is which?





OK, time to embarrass myself horribly:
raddevus wrote: Troll 3: Either we are all knaves or at least one of us is a knight.
That statement cannot possibly be false. Therefore, Troll 3 is a knight.
raddevus wrote: Troll 2: Troll 1 is lying.
If that were true, then Troll 1 would be a knave, and Troll 2 would be a knight. But that would mean that Troll 1's statement was true.
Therefore, Troll 2 is a knave. Which means that Troll 1 is not lying, making him a knight.
The fact that there are exactly two knights doesn't affect the truth of Troll 1's statement, since it was a logical implication, which only applies in one direction. If the lefthand side is false, then the statement is true regardless of the state of the righthand side.
On a related note, there's a simple solution to this drinking game: Saturday Morning Breakfast Cereal  Drink[^]
"These people looked deep within my soul and assigned me a number based on the order in which I joined."
 Homer





I'm very glad that you took a shot at it and felt uncertain about the answer, because I feel the same way.
Unfortunately, I didn't find the answer in the book.
Here's my logic though.
I thought Troll 2 was the only possibility for a knight.
I thought he was telling the truth that Troll 1 (a knave) was lying.
Troll 3 has to be lying because if all 3 were knaves, they'd all be lying and since Troll 1 and Troll 2 are in conflict with each other, one of them has to be telling the truth.
But, I cannot exactly rectify part 2 of Troll 3's statement (at least one of us is a knight).
But, I am entirely unsure about everything.
Is your answer correct? I will look it over again and see if it all seems to reconcile.
Thanks again for putting yourself out there. Maybe someone will come along who is confident of the answer.
Reposting riddle for reference:
Quote: While walking through a fictional forest, you encounter three trolls guarding a bridge. Each is either a knight, who always tells the truth, or a knave, who always lies. The trolls will not let you pass until you correctly identify each as either a knight or a knave. Each troll makes a single statement:
Troll 1: If I am a knave, then there are exactly two knights here.
Troll 2: Troll 1 is lying.
Troll 3: Either we are all knaves or at least one of us is a knight.
Which troll is which?
Edit
Solution 2 from me
as I read it again, I think that Troll 1 and Troll 3 are knights.
I think Troll 2 is lying that Troll 1 is lying so Troll 1 is a knight.
"My intellect is quite dizzying!!! Wait til I get started!!" ala Princess Bride. Princess Bride  "Truly, you have a dizzying intellect."  YouTube[^]
Then Troll 3 could be telling the truth that "at least one of us is a knight" (since there are two knights.
So
Troll 1 Knight
Troll 2 Knave
Troll 3 Knight
I've probably embarrassed myself more.
modified 24Jul19 17:21pm.





raddevus wrote: Troll 3: Either we are all knaves or at least one of us is a knight.
There's literally no way that statement could be a lie.
(Unless the setup of the riddle was a lie, and they're not all either knights or knaves. But that would be too devious.)
"These people looked deep within my soul and assigned me a number based on the order in which I joined."
 Homer





Richard Deeming wrote: There's literally no way that statement could be a lie.
Very good point! You are right. Very interesting part of the puzzle.





Nice, but one gripe I have is that you have to scroll to the top of the page to go to the next or previous topic. Bogus, man!
/ravi





Ravi Bhavnani wrote: Nice, but one gripe
It woudn't be the Internet,
1. if it wasn't free
2. there were no gripes about the free stuff






Hi All,
The new coffee machines at work are odd. Bean to cup! you get the usual buzz word coffees and the Decaf versions including Espresso decaf , why, I always thought Espresso was a quick short I need to be awake NOW, this second. A decaf version I think would concentrated coffee taste and thats it. I suppose it's offered as it used as base for the other drinks...but why?





I guess caffeine is addictive, and can lead to nasty side effects when abused of. Some like to get the placebo effect from it, though. Metabolisation time[^] being 45 minutes, the "NOW" effect does come from your brain, which reacts to the fact that it now knows that you have drank a coffee.
enum HumanBool { Yes, No, Maybe, Perhaps, Probably, ProbablyNot, MostLikely, MostUnlikely, HellYes, HellNo, Wtf }





phil.o wrote: which reacts to the fact that it now knows that you have drank a coffee.
No, no  it wakes up in anticipation!
Sent from my Amstrad PC 1640
Never throw anything away, Griff
Bad command or file name. Bad, bad command! Sit! Stay! Staaaay...
AntiTwitter: @DalekDave is now a follower!





Like a diesel engine?
enum HumanBool { Yes, No, Maybe, Perhaps, Probably, ProbablyNot, MostLikely, MostUnlikely, HellYes, HellNo, Wtf }





I just drank a large strong cup of proper (not decaffeinated) coffee, but I still feel as dozy and stupid as before.





MAybe you just dreamt you drank it.
Have another to check.
Sent from my Amstrad PC 1640
Never throw anything away, Griff
Bad command or file name. Bad, bad command! Sit! Stay! Staaaay...
AntiTwitter: @DalekDave is now a follower!





Tolerance to caffeine can build up. I have friends who can drink coffee and go to sleep after it. I'd be awake, staring at the ceiling.





Well I have been drinking it for over 60 years.





I am one of those that started drinking Coffee at age 8 about 62 years ago. I can have a cup before bed and fall asleep with no trouble.
I attribute this more to a clear conscience and a pure heart than to tolerance to caffeine...
Only the good die young  Billy Joel. What happened to me?





Oh. So we could serve him a tea made from my worn socks and it would work as long as he believes that this is coffee? Great. I think I will not wash socks anymore. Instead I'm going to shredder them and fill them into capsules, which I sell just under the current gold price.
I have lived with several Zen masters  all of them were cats.
His last invention was an evil Lasagna. It didn't kill anyone, and it actually tasted pretty good.





I don't think the worn sock thing is going to work; anyone may throw up, preventing the placebo effect.
enum HumanBool { Yes, No, Maybe, Perhaps, Probably, ProbablyNot, MostLikely, MostUnlikely, HellYes, HellNo, Wtf }





Never understood it either. And yet it is common...
GCS d(d+) s/++ a C++++ U+++ P L+@ E W++ N+ o+ K w+++ O? M V? PS+ PE Y+ PGP t+ 5? X R+++ tv b+(+++) DI+++ D++ G e++ h r+++ y+++* Weapons extension: ma k++ F+2 X





Dear Bob! I thought it was an aberration of these machines....





My former boss ordered a change of the vending machine because he wanted the decaf.
Poor man, he slept almost the whole 8 hours (well, he used to leave after 5 or 6) of work, if he drank coffee it would ruin his sleep schedule.
GCS d(d+) s/++ a C++++ U+++ P L+@ E W++ N+ o+ K w+++ O? M V? PS+ PE Y+ PGP t+ 5? X R+++ tv b+(+++) DI+++ D++ G e++ h r+++ y+++* Weapons extension: ma k++ F+2 X





Would you be interested in buying some of my special blend capsules? Not decaf, honestly.
I have lived with several Zen masters  all of them were cats.
His last invention was an evil Lasagna. It didn't kill anyone, and it actually tasted pretty good.





No, but I can ask to some of my former coworkers. Coffee and hygiene in that place were truly horrible, it would be an improvement.
GCS d(d+) s/++ a C++++ U+++ P L+@ E W++ N+ o+ K w+++ O? M V? PS+ PE Y+ PGP t+ 5? X R+++ tv b+(+++) DI+++ D++ G e++ h r+++ y+++* Weapons extension: ma k++ F+2 X





Have one of those machines in the apartment building I live.
About a year ago I started to drink espresso slowly with a cube of raw sugar on the side for afterwards while taking a break during the day. Picked it up from my little brother and sisterinlaw. According to them that is the proper Italian way to drink it. While I still roll my eyes when I think about it, they do spend a few weeks there near Bologna every couple of years. Her family has been doing it for a few decades now.
That said, I'd never trust espresso out of a super fancy Keurig to actually be any good.




