Logic riddles

[Alpha]

I have some logic riddles.
Lets start with a nice one, that starts like this:
Two hunters Mr. A and Mr. B meet in Mr. A's house.
Mr. B: Oh, what a beautiful carpet you have.
Mr. A: Yeah, it is made of bear's fur.
Mr. B: How did you hunt the bear?
Mr. A: Well, (Light his pipe :)), I started from the camp, and went 1 mile to the south, There I saw the bear.
I followed him exactly 1 mile to the east. There I shot him. From this point I returned to the camp, exactly 1 mile to the north.
The question is what is the color of the bear?

[blinc]

ummm... erm... uhh

Are you by chance a friend of Thespians? {toothy}

*grumble*
Need me a Big bottle of aspirin.
Big, big bottle yep.
These young whippersnappers tormenting us poor old fogeys
*mumble*

Will you give us clues? Like was it wintertime? or does it matter if it was winter or not?

By the way Alpha, it's nice to meet you and glad to see you're posting already! :)

[KatCo]

I believe the bear would be white as in a Polar Bear, to go in a circle and end up back where you were would mean that you would have to be at the pole. Just a guess....

[Alpha]

Excellent KatCo!
The Answer is correct.
The only place on earth in which one can go south, east and then north and return to his starting position is the North Pole, because the longitudes meet there.
So this is the only answer, or is it?
Ignore the bear in the story, is there another place on earth where it is possible to go the same path and return to the starting position?
Be careful this is a very difficult question!

[KatCo]

humm....other than the south pole I would assume???
the "same path" has me thinking.....no answers as of yet.
tis a tough one......I'll keep a pondering.

[Alpha]

The South Pole can't be the answer, because you cannot go southward from it (In the story he went 1 mile south, east and then north).
As a matter of fact if you are standing exactly on the South Pole you can go only northward (At least in your first step...:))
There is a solution, but it is quite hard to come up with it.

[blinc]

South, then east and then North... the equator could be the other answer couldn't it?

You know if you traveled at angles you could always end up back at the same point. Head S at an angle like this / then straight east _ and then the reverse angle back north again... you could make a giant triangle... but I assume you mean straight south, huh? If it's not the equator, I'm really stumped over here! Great logic riddle Alpha!

[Alpha]

The equator is not the answer.
As a matter of fact, the equator is where the coordinates are nearest to cartesian coordinates (where the axis are straight and in 90 degrees) and therefore the hunter would finish his journy in the farthest point from his origin. That distance would be of course exactly 1 mile.

As I told you, this is a tough question.
If nobody solves it I will give the answer in few days.
Good luck.
{devil}

[blinc]

Arrgh! You've stumped me Alpha - thought the equator would work because if you were standing at the very edge of the equator "line" and tried it, was thinking you would end up back on the equator line, but of course, one mile to the west of where you started... back on the line? well dang, I sure've mulled this one over - haven't given up though!

[blinc]

*pulls out tufts of hair in frustration*

AAAARRRGGGGH!!

[Alpha]

The question was really hard.
As a matter of fact there is not only 1 place like this, but infinite number of places.

Before I reveal the solution I will remind you what does going south, east, north and south mean:
Going north and south is simply going up and down on the longitudes. Going west and east is going right or left on the latitudes.

The longitudes are going from the South Pole up to the North Pole.
The latitudes are parallel to each other and go around the globe in circles that are small around the poles and largest near the equator.

Since the latitudes are small near the poles, one can find a latitude near the South Pole that is 1 mile long in circumference. This latitude would be about 250 yards from the pole (1 mile/2/Pi).

If one stands on this latitude and walks 1 mile to the east, he will go around the globe (that is easy near the poles) and return to his starting position.

If one stands 1 mile to the north of that latitude he would be able to go 1 mile to the south, 1 mile to the east, and then 1 mile to the north and return to his starting position.

The solution, therefore, is all the places on the latitude that is about 1 mile and 250 yards north of the South Pole.

Of course he won't find any polar bears there. The polar bears are only in the North Pole. In the South Pole there are penguins, and not so far inland ... :)

This was a long explanation, but not the end of the logic riddle.
The next part of the riddle, as you could guess is...
Find another point on the globe that one can go fron it 1 mile south, east, and north and return to his starting position. {devil}

[blinc]

*stares*
*twitch*
*stares some more*
*drool*
*twitch*

[Anat]

He is wicked isn't he?

That's okay blinc, I can NEVER find the answer to his logic riddles. My poor little brain is not up to it.

In fact I couldn't follow the explanation very well and he had to show me (we used an orange as a model of the earth :) - then we ate the whole planet :)). I finally got it, but I couldn't repeat the explanation myself :D

[thespian]

"Okay - Lights! Camera! Action!"

The thespian has arrived!

Listen Alpha all this talk about longitude and latitude has got me going round in circles ( Ha! Circles - get it? )

I know of another place where I walk a lot more than a mile in the same direction and always end off at the same place I started from. {mario} It's at the Gym around the corner.
"Okay everybody, let's get those Treadmills fired up. Come on, all together now - {mm} We're on the Road to Nowhere ..." {mm}

[Anat]

He's not home yet, but does all that song and dance means you're actually trying to solve this?

Just curious...{rolleyes}

[KatCo]

Hi, This is not Kathy, but her son James. I have a question about the way you dervived your answer. If you were to travel 1 mile south from base camp then one mile east to catch the bear, then head right back to base camp, I would assume that the hunter took the shortest path home. That path would be the tangent line on a right angle triangle. According to the Pythagorean Theorm A^2+B^2=C^2 where C= to the tangent line. If he traveled 1 mile south (in a strait line) the 1 Mile east, these make up the legs of the triangle or A and B. Therefore 1^2+1^2=C^2 solving for C give 1.414 miles as the tangent line. You said that he traveled back north 1 mile, this would put him 0.414 miles shy of reaching base camp. Unless of course if the hunter was walking along the circunference (which is not walking in a strait line), then he would always be 1 mile south of base camp. Also, if one were to go out 250 yds and go east 1 mile, this person did not traverse the entire globe. In order to do that, this person would have walk around the "widest" part of the earth because that is where the maximum radius of the sphere exists. According to the diffination of Volume using Riemann sum: V= the limit as n approches infiniy of the sum of A(xi) change of x starting at i=1. therefore as n approches infinity you are cutting a sphere into multiple "plates" with smaller and smaller radi and therefore smaller and smaller circumferences. These plates are like the lattitudes on the earth. The maximum and true radius and circumference of a sphere is the widest part of the sphere, and in the earth's case, the equator.

[blinc]

{awed smiley}

Ok, where'd that bottle of aspirin go? Gosh, it's amazing to see the smarts some of the people here on the forum have. I'm flabbergasted. Amazed. In awe and in pain! {toothy}

[KatCo]

It is James again. As long as you can define any point on the earth as (0,1) on the rectangular Cartesian Cooradent System which in our case can be defined as absolute north, then you can go 1 mile south, east then north and end where you began. So I think that if you did this, then any point on the earth is the answer as long as you define the starting point as (0,1) because 90 degrees from this point would be the equator. However, I do not think this is what you are looking for.

[jamesglewisf]

James,

If you would like your own member name, I can create one for you. All you have to do is send me your email address and desired member name. I'll give you a temporary password that you can change to your own. The email address cannot be the same as your mom's.

If you are under 13 years of age, I'll first have to send you a form that your mom can fill out and either fax or snail mail to me.

Just click on my mail icon on this post.

[Alpha]

Hello James
You wrote really nicely.

You have wisely mention Pythagoras theorem, Nether the less, this theorem holds only in Euclidian geometry. This geometry assumes a flat world.
When we deal with a different case, like walking on earth, we sometimes can't use this theorem.

The same goes with Cartesian coordinate system. There are plenty other coordinate systems one can use and in many cases the decision of the coordinate system is crucial for an easy solution.

For example, one can draw a triangle of 270 deg on earth, where on side of the triangle would be the Greenwich longitude from the equator to the north pole, another side would be a longitude in time zone Greenwich + 8 hr from the equator to the north pole and the first side would be on the equator.

By the way, this is one of the methods used to determine the radius of the earth.

When our hunter finished his 1 mile south and 1 mile east, the shortest path to his camp (if he doesn't dig...:)) was going 1 mile to the north, and not the sqrt(2)*1 mile.

When the hunter went eastward he walked on a latitude (this is the definition of east-west). The latitudes really get smaller and smaller when you get closer to the poles - try painting them on a ball.
Near the poles you can find latitudes that are as small as you want, including latitude that is 1 mile long.

[KatCo]

Alpha, After you explained that i can now conceptually "see" what you are saying. The 1 mile east is in fact walking on the circumference! It is hard for me to see 3D space in 2 dimensions, but what you said makes sence! However, I am still stumped on the last part of the logic riddle. And besides, whos to say the world is round LOL!

[thespian]

Originally posted by Anat
... but does all that song and dance means you're actually trying to solve this?Not really, it was just my valiant attempt at humour. Somewhat warped, it never quite comes out the way it was meant.

No, I leave mathematical type riddles to you geniusses out there. I prefer the "trick" type. Maybe I need to seek professional help?

Have a great weekend everyone!

[blinc]

oh boy, this one was so tough I wasn't going to post it. But, after reading the conversation before this post, have the feeling nothing is too tough for you guys! So, here goes:

Alice, Ben, Charlie, David, and Ed entered a contest to guess how many jelly beans are in a jar. Alice guessed 30, Ben guessed 28, Charlie guessed 29, David guessed 25, and Ed guessed 26. Two were off the mark by 1, one was wrong by 4, and one by 3. But one was correct. How many jelly beans are in the jar at the store?

[KatCo]

OK....I believe that Charlie has the correct number of jelly beans at 29.

Ben was 1 under at 28

Alice was 1 over at 30

Ed was off by 3 at 26

and David was 4 off at 25....

And this is Kathy, James is now going to post under Dk Tagh...these logic puzzles are turning into a family thing....keeps us out of trouble...ha ha ha

[blinc]

Oookaaay, *ahem* so maybe it's only tough to certain people on the forum? {toothy}

Yes, Katco! Way to go, you got it on the first try!

[KatCo]

I think that I have the answer...

I believe that Charlie has guessed the correct amount at 29.

Ben was under by one at 28.

Alice was over by one at 30.

Ed was under by 3 at 26.

And David was under by 4 at 25.

This is Kathy by the way, my son James will be posting under Dk Tagh...these logic puzzles are keeping us busy!! It's becoming a family thing....much better than some of the TV programs that are on!!!

[blinc]

I saw the first post Katco... couldn't you see it? I answered your post. Can you see your post now? If not, just let us know and we'll try to find out what the problem is.

[blinc]

Well, since this is the thread for the more logically minded people, I found one you all might enjoy. These kind make my head hurt to figure out, but I sure like to post 'em! {toothy}

At a recent visit to the reptile house at the local zoo, I counted a total of 27 heads and 70 feet. I was counting snakes, lizards, and people. I know that there were exactly twice as many lizards as people. How many snakes did I count?

[kezzer]

18 snakes?
This is my first time venturing into this section. I used to be very good at logic puzzles, but I think in age I lost my own logic!!

[blinc]

Oooh! Another fun person to tortur... to uh, play with, yes! That's it, always great to see another person join in the fun!

This is the best part of riddles, you get to sit back, gloat a little and say:

Nope! It's not 18. {toothy}

[Debby]

I'm not even going to attemp this one, but I have to say, after reading this entire thread...I feel a bit depressed...LOL

I always thought I was a semi-intelligent person...but after reading Alpha and James's posts...about the bear...I feel completely ignorant....I was so lost on all that stuff you guys were talking about that my head was swimming...I need your asprin Blinc!! Gasp!! (I am in awe!!!!!!!)

[kezzer]

16 snakes, 3 lizards, 8 people???? This is killin me!

[thespian]

I love these riddles - please let's have some more.

By my calculation there were 6 Snakes (7 People and 14 Lizards). I wonder how many of the people were really snakes?

[blinc]

Aaagh! Thespian got it on the first go. Unbelievable! Way to go Thespian!!!

Geesh, can I borrow that dunce cap back Debby? {dunce} There's no way I could have got this one that fast (maybe in a couple weeks). Let me go see if I can find another. harder, much harder riddle for ya! {toothy}

[blinc]

Ooh, this one's a stinker!

With a little detective work and deep thought, the following facts can make an accurate statement. Can you figure it out?
11 is a race horse
12 is 12
1111 race
12112

[kezzer]

*drool*
*twitch*
*drool*
I'm not even going to attempt that!

[thespian]

I know you're gonna kill me but here goes ....

One-One is a race horse,
Tu-Tu is one too.
One-One won one race,
Tu-Tu won one too.

Okay, let's have a litle fun. How many times can you say the word "Rite" consecutively in the same sentence and have it still make sense? I am of course referring to the phonetic pronunciation of the word. For example one might say "That is the right rite", meaning that someone has executed the correct procedure for example.

I have get to six times, any offers?

[blinc]

Well, I'll be.... Thespian you worked out a different answer then the one that was given! You sure do get credit for it anyways, since your answer works too! two! to! {toothy}

Here's what the other answer was:

11 is a race horse (Eleven is a race horse)
12 is 12 (Twelve is one too)
1111 race (Eleven won one race)
12112 (Twelve won one two)

The rites...ooh, I can come up with five.

To find the right rite, look just to the right... oh darn, forgot how I had it in my head before I began this post! Arrgh! I'll be back for sure!

[DkTahg]

A combinded effort of KatCo and Dk Tagh (James and Kathy) came up with this, it's probably not what you are looking for but here goes...

Write right rites right; right rites are right.

Meaning if you write proper rites correctly then right rites are correct....

Wow....words are not our forte, but we sure do enjoy this forum...thanks for a good family time.

[DkTahg]

Here is a new puzzle for you all to ponder...

Gnomes & Goblins

3 Gnomes and 3 Goblins arrive at a river bank, and they all wish to cross onto the other side.

Fortunately, there is a boat, but unfortunately, the boat can only hold 2 creatures at one time.

Also, there is another problem, Goblins are vicious creatures, and whenever there are more Goblins than Gnomes on one side of the river, the Goblins will immediately attack the Gnomes and eat them up! (Oh my!)

Consequently, you should be certain that you never leave more Goblins than Gnomes on either river bank. How can this problem be solved? It must be added that the Goblins, though vicious, can be trusted to bring the boat back.

Hint: it is necessary to temporarily work "away" from the goal.
This is pretty tough {devil}

[blinc]

Oh boy, this is a toughy... I need to print this one out and get a pencil!

Great riddle DkTahg! :)

[blinc]

I've got this one printed out DkTahg... don't give the answer yet!! Just haven't had time, what with the holidays, to sit down and give it a go. Sure it's the same for the others too. :)

[Karenluvs6]

why can't all the gnomes and goblins just take the bridge across the stupid river bank!

[Karenluvs6]

I'm just kidding, of course!
I actually know this one....I have heard many variations of this joke.

[DkTahg]

No problem Blinc! The holidays always seem to keep keep everyone running around in circles {headspin}....

I would have to agree with you, the bridge would be the "logical" way to go....I was also taught that I should try to get along with "others" but I guess these guys "lack" any parental guidance {eek2}, what is this world coming to??? Those little Gnomes and Goblins need to learn to co-operate with one another.

I guess until they can learn some interpersonal skills, that boat is going to have to make a lot of stops!

Yeah there are a lot of "variations" of this riddle, this is one of the most "complex" one's I've seen. Sorry guys {evil}

[blinc]

Ok, I think I've got this worked out...

Here goes. You start with 3 on each side.
We'll put the Goblins on the left side, marked with X's and
Gnomes on the right bank marked with 0's

X......0
X......0
X......0

You put one Goblin in the boat to be the "driver", which leaves two on the left bank. The driver goblin picks up one Goblin, takes him to the Gnome side. Picks up one Gnome and takes him to the Goblin side. That leaves one Goblin on each bank and one in the boat.

X....X(the one in the boat, or middle).....X
0..........................................0
...........................................0

Pick up another Gnome, bring him back to the goblin side.

X.....boat X....X
0...............0
0
And another Gnome
X... boat X... X
0
0
0
Make one more trip to pick up the last Goblin, take them to the Gnome side. On the last trip back both Goblins get out and they're switched.

Why do I think I've missed something here? Does the Goblin in the boat count as being on a side when he's actually in the boat? cuz if he does, it's back to the drawing board big time! {toothy} Although if it is... I've got the ending worked out, but have to rework the beginning.

[DkTahg]

You started wrong, but it was a logical start!
All of the gnomes and goblins are on the same side of the river to begin with.

so..
| |
X | r | you must get all of the gnomes and goblins
X | i | onto this side of the river. I will give you
X | v | another chance to make your brain hurt.
O | e | You better stock up on the asprin! I will
O | r | give you some hints but if you don't want to
O | | use them, don't read them.
| | 1st. It sometimes seems that you are moving
bacwards when in fact, you're moving foward.
2nd. The boat will make a total of 11 trips
across the river.
Third, dont eat yellow snow {eek2}.

[blinc]

Thought it worked out too easy... I read the puzzle wrong. Arrgh! Ok, ok... back to my little piece of paper and big bottle-o-aspirin.

This is a fun riddle DkTahg! :)
ummm... is it ok if we go over 11 trips? cuz, I may very well end up with 38 before all's said and done. hee hee!

[blinc]

ok, definitely need to know this before I hurt myself. Is whoever in the boat considered neutral? Or if he's on one side while dropping someone off, he's considered to be "on that side" in general.

[Debby]

good question, i was wondering that also.

[blinc]

Ok, he isn't neutral. The hubby and I sacrificed a refrigerator magnet into six parts and worked this bad boy out. {toothy} Here's what we came up with:

Put two goblins in the boat
Drop one goblin off on the other side
Come back, pick up the other goblin
Take the 2nd goblin across to the other side
Goblin brings boat back, two gnomes get in
They go over and both get out on the other side
The boat is then given to a Gnome and a Goblin
They come back across, put two gnomes in the boat
go over to the other side and they both get out of the boat
The boat is given to a goblin
The goblin goes back and picks up a goblin
brings him over drops him off
A goblin goes back to get the last goblin and they both get out.

Eleven moves. Yup. This was a great riddle. For awhile we didn't think there was a way to do it, but one piece fell into place and we worked it out from there.

Great riddle DkTahg! Lots of fun! :)

[thespian]

Originally posted by blinc
Yup. This was a great riddle. For awhile we didn't think there was a way to do it, but one piece fell into place and we worked it out from there.

Great riddle DkTahg! Lots of fun! :)


I second those thoughts. My three 10c coins and three paperclips have proven the theory! Well done Mr. and Mrs. blinc (and fridge magnet)!

[blinc]

Yup, that poor ol' magnet saved the day. Thank goodness they make them out of cuttable material nowdays, or we would still be working on that one! That was good fun. :)

Here's another logic riddle, but not mathematical in nature, rather just using the ol' noggin to find a solution.

You have a ping pong ball that fell down a curvy, long, narrow hole in the ground. You want to get it out but you can't stick your hand in it won't reach or fit. A string won't work because you won't be able to attatch it.
What can you do to get it out without digging a hole?
(No, you can't send a ferrett or other pet down to fetch it. :) )

[DkTahg]

Wow, you did it! My hats off to you. {einstein}

[DkTahg]

Fill the hole with water. Ping pong balls are boyent in water so it will float! Or try a syphon to suck it out. If that didn't by a new one. he he.

[blinc]

{dude} My hats off to you DkTahg! You got it on the first go! Fill the hole with water was the official answer! :)

[blinc]

Ok, who's got the next logic riddle? I've got a fresh bottle of aspirin just begging to be opended! {toothy}

[KatCo]

When the day after tomorrow is yesterday, today will be as far from Wednesday as today was from Wednesday when the day before yesterday was tomorrow. What is the day after this day?

[TTP]

I'm thinking it is going to end in y... {toothy}

[KatCo]

Yep, I can guarantee, most certainly that it will end in a "Y" tee he he...;)

[blinc]

Don't ask me how, because thoughts were just bumping around in the ol' noggin... but is it Saturday? Wednesday.. two days further away and one more day past the 2nd day... is pretty much how it was jumbling around in there.

*sigh* not even close, am I? {dizzy}

[KatCo]

Blinc, your answer is not correct, but the fact that you are trying is impressive. I must admit that this one gave me "fits" :8| when I first saw it. It is confusing as all get out (at least it was for me).

OK, I am hoping that this will help you, the day in which you begin is Wednesday.

Then you "rename" the days of the week: Making Wednesday today.

Then you do substitutions.

I will reveal the entire "workings" to you later if you want, but thought I would give you one more "attempt" at it.

[DkTahg]

Blinc, Im amazed that you are attempting to solve this nasty little riddle. It gave me such a head ache, and it still has not gone away {frusty}. However, I know the answer so i can't give it away. Again good job, and good luck :)!

[blinc]

Ah sure, sit over there and gloat DkTahg! {toothy} This one's gonna hurt my brain, isn't it? *grin*

Ok... start with Wednesday.

When the day after tomorrow is yesterday, today will be as far from Wednesday as today was from Wednesday when the day before yesterday was tomorrow. What is the day after this day?

When the day after tomorrow, is yesterday... (today was Wed.), so day after tomorrow would be Tuesday?
New Today (tuesday) would be as far from Wed. (one day back) as today (tuesday) was from Wednesday (one day) when the day before yesterday (Sunday?) was tomorrow (Monday?). The day after this would be...uhm Tuesday?

{in pain smiley} excuse me please, while I go get that new bottle of aspirin. I got lost in the second part and not sure where I was... Oh yes, now I remember... in pain!! {dizzy}

[DkTahg]

Ok time to solve:

This is the best way that I can explain it:

1. When the day after tomorrow is yesterday
2. today will be as far from Wednesday
3. as today was as from from Wednesday
4. when the day before yesterday was tomorrow
5. What is the day after this day?

The problem is telling you which day to begin on: Wednesday:
that makes:

Monday: the day before yesterday
Tuesday: yesterday
Wednesday: today
Thursday: tomorrow
Friday: the day after tomorrow

Now substitute:
1. When the day after tomorrow (Friday) is yesterday (Thursday).
2. today () will be as far from Wednesday
3. as today () was as far from Wednesday
4. When the day before yesterday (Monday) was tomorrow (Tuesday).

Today will be Wednesday because it is the day which falls between (1. Thursday) and (4. Tuesday).

5. What is the day after this day? AND the answer is:
THURSDAY.

Now that that is clear as mud, I hope that I didn't cause any aneurisms! UG, :8| Next time I won't post such a brain drain one, I promise.

[KatCo]

Sorry, I forgot to change the Username, that really was me Katco that posted the answer, besides I'm not sure Dk Tagh could have done such a fine of job of explaining it as I did...he he he...little jab there at my son.{toothy}

[DkTahg]

Gosh, I dont know what to say to that. Grandma understands me. :( I think i'm going to go live with her now. Lets see what you'll think of that hehe! No, actually I will always love you no matter what. {love}

[DkTahg]

I love my Mom but not that much!

[blinc]

{toothy} Oh, I'm just a giggling over here. It's such a relief seeing someone else is having "one of those" days. *grin*

Thank you, thank you, thank you KatCo for giving the answer. I don't think I would have survived another stab at that one. whew!

[Debby]

I also got a kick out of you posting the wrong smiley!! Too cute :)

[PsalmReader]

Well I think I found where DkTahg has been. :) I said "Howdy", but I'd never seen a post. I hadn't checked out this forum and the threads here before. Cool stuff. :)

[Debby]

I have a question....not sure this is the right place to post it....but give it a shot, okay?

can anyone name the four member rock group in which one member was assasinated, and the other three died of natural causes?

It sounds like trivia...but really it is a riddle, I think.

[KatCo]

Um..........Uh........

Scratching head here.......

Yipes, havn't got a clue as of yet. Was era was this group? Can you tell us or will that give it away?

Ug....{confused}

[DkTahg]

Brain...shorting...out. bzzp..bzzp.
Can you clairify something for me?
Are you talking about a group of rocks, or a group of musicians? Also, when you speak of group, do you mean as "all in one," or as in "having like properties?"

[Debby]

Actually yes...it is a group of rocks...but that pretty much gives it away now! Good thinking, DKTahg! Now I'm sure you will get it!!

[thespian]

I Got It! And I aint even an American! It's that mountain thingy with the faces on it. Mt Rushmore I think?

[blinc]

Ooh, I bet you are right! That was a great guess Thespian, I wasn't even going in the right direction on that one. :)

[DkTahg]

Sounds good, you got my "chad."

[blinc]

As soon as we hear from Debby if you got the answer right Thespian, we'll go ahead and close this thread since it's getting soooo long.

Feel free to start another one in the meantime!

I bet you've got the answer.. it just fits so well, can't think of what else it could be! :)

[Debby]

Yes!!! That was the right answer!!!! :)

[blinc]

Well there ya go Thespian! Congratulations!! {toothy}