DS: In a certain sport, teams receive 3 points for each win

[NL]

In a certain sport, teams receive 3 points for each win, 1 point for each draw, and no points for losses. In a five-team tournament in this sport, in which each of teams G, H, J, K, and L played each other team exactly once, did team L finish the tournament with the highest point total?"¨

"¨(1) Team L finished with 8 points."¨
"¨(2) The sum of all five teams’ point totals for the tournament was 23 points.


Dear Ron,

I feel so dump with this problem, especially when it comes to combining both statements.

Here is my thought process:

Conditions:
5 teams: G-H-J-K-L, each plays each other team once.
- win: 3 points
- draw: 1 point
- lose: 0 point
Goal: whether L has the highest point?


(1) "L finished with 8 points"
- Specifying the statement:
+ 5 teams: "G-H-J-K-L, each plays each other team once" means: there are 5C2=10 games, and each team plays 4 games.
+ "L finished with 8 points" means L wins 2 games and draws 2 games (only this case is possible)

- Whether L has the highest point?
YES: in case no other team wins 2 plays, so the second-highest team can have: 3+1+1+1 = 6 points.
NO: in case one team, e.g H, loses to L, but wins 3 remaining games. So the point H can get is 9.
→ Insufficient!

(2) The sum of all five teams’ point totals for the tournament was 23 points.

3W + 1D + 0L = 23
Although we have only 1 value of for W and D: W=7, D=2
But too many cases can happen.
E.g:
- YES: L wins 4 games, and the second-highest team wins 3 games & draws 1 game
- NO: L doesn't win any game, and 7 winning-rewards distribute to 3 remaining teams.
(in general, W & D can run from 1 to 10, but will exclude some value in the relationship with other variables)

(1)&(2): I want to checked to see whether 2 cases in (1) can happen when total points was 23.
3W + 1D + 0L = 23 and L gets 8 points

Case "YES": Ehem... I got stuck here! Because I thought too many cases can happen. (I ended up with E in the CAT)

Please help me with how I can use common sense to "see through" this step.
The answer is C.

[RonPurewal]

You're assuming too much freedom in statement 2.

Remember, each team plays each other team exactly once. So, there are exactly 10 games played in the tournament. (You can use combinatorics to figure this out if you want; I think it's easier just to list out all the games.)

Also, don't forget"”If one team wins a game, the other team has to lose it! If there's a draw, then that draw counts for 2 teams!

Think about the total of 23 points in light of all these facts. It determines more than you think it does.

[NL]

You're assuming too much freedom in statement 2.

Remember, each team plays each other team exactly once. So, there are exactly 10 games played in the tournament. (You can use combinatorics to figure this out if you want; I think it's easier just to list out all the games.)

Also, don't forget"”If one team wins a game, the other team has to lose it! If there's a draw, then that draw counts for 2 teams!

Think about the total of 23 points in light of all these facts. It determines more than you think it does.

Thanks Ron! I really like your explanation. It made my brain crawl!
The important information I missed was me-called hidden conditions, conditions supposed to get from "real world situations"- (Ron’s words).

It shows up here:

3W + 1D + 0L = 23

At that time I assumed that W+D+L =10 (games)

There are 2 things wrong in this equation:
- 1st: L can not be here, because W and L will stay in the same games, as you pointed out.
- 2nd: "1D" is incorrect, because 1 draw game would bring 2 points for 2 teams.

→ So the correct equations are:
3W + 2D =23
W + D = 10
→ W =3

So if L wins 2 games and ties 2 games, the possible second-highest point would be 5. Sufficient!

Is any thing else wrong in my reasoning?
(The question is still hard for me. I’m pretty sure this question was created by Ron Purewal)

[NL]

I just have a little comment: the background's color of the quote's part should be little different, so that reply parts and quote parts can be distinguished easily. Anything else is perfect (for now)

[RonPurewal]

I just have a little comment: the background's color of the quote's part should be little different, so that reply parts and quote parts can be distinguished easily. Anything else is perfect (for now)

You can use the vertical lines as a guide. I see a distinct, easily visible box around each quote; if the box appears faint on your screen, you may want to increase the "contrast" setting on your monitor (or decrease the brightness).

There is a LOT of quoted text on this forum. Changing the background color to anything but white, gray, or light beige would get very irritating, very quickly. (There's lots of research on this, but, rather than looking all that up, just find a blog with a colored or shaded background, and see how long you can read it before you start getting headaches.)

[RonPurewal]

And, yes, I wrote the question.

[NL]

Changing the background color to anything but white, gray, or light beige would get very irritating, very quickly.

He, that’s right: if the background of quotes were more gray, people would quickly recognize where a quote ends. Usually, people just take a glace of a quote to see what point to which the Instructors respond and want to quickly jump to explanations, especially to a topic containing lots of LONG quotes and explanations.

(There's lots of research on this, but, rather than looking all that up, just find a blog with a colored or shaded background, and see how long you can read it before you start getting headaches.)

Oh, I see. Look at the big red bar bearing "quote" right here: it seems like the owners of the elegant house, although very welcoming, always highly take precaution against thieves "hey, dare you to come here! Will give you taste this (stick)"

And, yes, I wrote the question.

See. I told you :))

[RonPurewal]

Ok. Well, I asked two other moderators, and everyone concurred that it's pretty easy to see where the quotes begin and end. So, let's kill that discussion. (If you genuinely find the borders of the quoted text hard to see, it may be a display issue, as suggested above.)

If you have anything else to post about the problem, fire away. Otherwise, let's consider this discussion closed. Thanks.

[NL]

He, last words here: "two other moderators" were not representative enough for viewers. Anyway, little "hair" is generally better than no hair. Thanks Ron!

[RonPurewal]

Hair?

[NL]

The discussion was closed, Sir!
(there is no reason to think that 2 entities in the 2 quotation marks are similar--hihi-- Just an analogy!)

[RonPurewal]

I still don't get it, but, ok, the discussion is closed.