## A Logical Meal

Four logicians were having dinner and discussing logical puzzles. After the main course, the waiter brought a large plate that contains 11 slices of cakes. Their debate was intense and in the process, they ended up eating all the cakes. Everyone had at least eaten one cake, and each of them is aware of that fact. Each logician knew how many cakes he ate, but they didn’t have the knowledge of how many cakes each of the others ate.

They had the following conversation to try to find out how many cakes the others ate:

Alex: Did you eat more cakes than I did, Ben?

Ben: I don’t know. Did you eat more cakes than I did, Charles?

Charles: I don’t know.

Danny: I got it!

Danny seemed to find out how many cakes each logician ate solely based on the conversation above. Can you also figure it out?

## Solution

I tried to give this puzzle to some of my friends who are good at math (a lot better than me) but they were all stumped. Funnily, I asked a waiter, who’s also a friend, and he was able to solve it.

The question looks complicated but it’s actually quite simple if you give it some thoughts.

Since Alex has asked, “Did you eat more cakes than I did, Ben?” it implied that Alex has not eaten 5 cakes or more. If Alex has eaten at least 5 cakes, he would not have asked Ben that question. Ben could not have eaten only one cake since if that’s the case, then he would have known that he hasn’t eaten more than Alex. Of course, Ben also didn’t eat 5 or more so that means that he either ate 2, 3, or 4 cakes. As an excellent logician, Charles has already figured this out. However, he was still not aware whether he ate more cakes than Ben. This means that Charles has eaten either 3 or 4 cakes.

Danny could only deduce how many cakes the others ate if he ate 5 cakes. If he ate, let’s say, 4 cakes, then there would be a lot more possibilities which would make it impossible for him to determine the answer. And the others must have eaten 1, 2 and 3 cakes to add up to 11.

In summary, we can conclude that Alex ate 1, Ben ate 2, Charles ate 3, and Danny ate 5.

My name Edmark M. Law. I work as a freelance writer, mainly writing about science and mathematics. I am an ardent hobbyist. I like to read, solve puzzles, play chess, make origami and play basketball. In addition, I dabble in magic, particularly card magic and other sleight-of-hand type magic. I live in Hong Kong. I blog at learnfunfacts.com. You can find me on Twitter @EdmarkMLaw and Facebook. My email is learnfunfacts@gmail.com
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### 44 Responses to A Logical Meal

1. Reblogged this on Site Title.

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2. psiberite says:

Great way to solve a puzzle, Anand Bose from India

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3. Sorry, but this has all the hallmarks of a Mensa puzzle and is an ample demonstration of the reason I no longer pay my membership fee! With all the real problems and issues that beset the world it is positively depressing to discover its supposedly most intelligent people sitting shrinking their heads for hours over cakes! Good puzzle, though!

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• Heh, when I was still in the US, some people suggested me to join MENSA. Of course, I ignored those suggestions. Bu5 on time, I saw a leqflet which said that there would be MENSA Conference near our area. To sate my curiosity, I asked a friend (who’s a MENSA member) to accompany me.

Long story short, the conference was pretty good overall and I got to meet a few interesting people. However, I was also disappointed. Several of the people I saw act like a socially awkward penguin. Their social interaction skills made my cringe. Also, several of them seem to use IQ level as a measure of worth of people. There were lots of talks like, “only people with an IQ of 150 or above would understand what I was saying”. They look down on people with “average” or “below average” IQ as if they can’t function properly in a society.

IQ level alone doesn’t define a person just like test scores alone can’t be used to measure a student’s intelligence. There are so many variables that can’t be easily “tested” and assessed after all.

Sorry for the mini-rant btw, and thanks for reading.

Also, I came up with this puzzle. It sounds too frivolous indeed. Hah, I still have a long way to go… 🙂

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4. I love this! I didn’t get it though, but fun piece. Thanks for this great post!

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5. natuurfreak says:

Puzzelen is niet mijn ding

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6. sydlogsdon says:

Hi Edmark, I’m glad you came along with your conundra. (deliberate error) I’ve been blogging for a while and trying to figure out what the stats are telling me. Today I blogged some things I thought no one would care about, just because they are what I’m up to this week, and got an unusually big response — for me. I had 18 views from 9 visitors and 9 likes. 6 views came from the USA, 5 from India, 3 from the UK, 2 from Romania, 1 from Ukraine, and 1 from Hong Kong. That’s a lot of data, and as usual I have no idea what it means because it could represent dozens of realities, depending on what assumptions I make. I’ve decided not to worry about it, and go eat some cake.

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• It’s hard to predict which post would become popular. This happened to me several times.

Anyway, the view from HK comes from me.

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7. Very confused about the number of cakes eaten by each. However, I do know that now I want cake as shown in the photo! 🙂

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8. Ooooh yes another great puzzle as always, love sharing these great stuff

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9. Tried this on my partner (retired maths teacher). Kept him puzzling for a while, but got the right solution. He also spotted the nice name association with A,B,C & D, which I have to confess, I hadn’t noticed. My mind obviously absorbed with that great image of cake!

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• Good. This problem seems hard but it’s pretty easy if you give it some thought. It’s also essential not to complicate the problem. That’s what my mathematicians have done – overthinking it too much.

The ABCD pattern is of course intemtional to make solving the problem. Though it’s understandable that you didn’t notice it because of the picture of the cakes 🙂

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10. dweezer19 says:

Who cares? I’m just mad I don’t get a puece of cake! 😜

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11. thelavainnah says:

I really enjoy the logic behind this cause it took me a couple of re-readings to get me understand it haha

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12. That was fun! I did it this way: if Ben does not know if he’s eaten more cakes than Alex, He has to have eaten more than one (at least two).Charles, being a logician knows that, but he does not know if he’s eaten more than Ben, si he’s eaten at least three. So, for Danny to know this, he’s eaten five.

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13. Lilly Moore says:

Amazing! I love logic puzzles like this.

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14. Graham says:

Some people just need to enjoy cake and not think about it too much. 😄

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• They enjoyed eating the cakes too much so they weren’t able to see how many cakes the others ate despite how observant they are. 🙂

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• Graham says:

I think my other half spends far too much time trying to ensure she gets at least as much cake as I do…ridiculous. She seems to enjoy it too. That is women and multitasking for you.

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• jfrels says:

I have two daughters and my policy with splitting cakes is:
Daughter A cuts cake in half.
Daughter B choses which half she gets.

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• Interesting.

Your daughters must very proficient at splitting cakes in half now :).

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15. tammy7711 says:

Haha🙌😂 Awesome post! I love it!

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16. Jess says:

Loved this question, I came up with 2-3-4-2… I understood how your explanation worked but not really sure why mine isn’t right too…?

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• Because if Danny ate 2 cakes, there would simply be too much possible combinations of answers for him to figure it out.

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• Jess says:

Oh right I see, I was trying to figure out how Danny knew but the question is actually aimed at the reader to figure out how many cakes everyone ate! Thanks for the clarification 🙂

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17. jfrels says:

Took me a few minutes, but I made myself work it out on paper before I read the solution.

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18. Jeff Rab says:

Great puzzle! I came up with 2 – 2 – 2 – 5. I based this on Alex’s question to Ben asking if he ate more “cakes” than him. Cakes is in the plural, not singular, which to me, indicates that Alex ate more than one. If any of the first three had eaten one cake, they would have known the answer. Thus, each had eaten two cakes (plural) and Danny ate five cakes….bad logic??? Or right answer???

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• Alex knew it was possible for Ben to have eaten more cakes than he did. Alex thus ate a maximum of 4 cakes. If he ate 5 cakes, Charles and Danny each ate at least 1, leaving a maximum of 4 for Ben. Alex’s possiblilities: 1,2,3,4

The other three logicians also knew everything said above. Ben says he doesn’t know if he ate more than Alex. This means he ate more than 1 cake, because he would’ve said “no” to Alex’s question if he ate 1. We are on the same page with this, right?

But then Ben asked Charles if he ate more. This signifies that there is a possibility of Charles having eaten more than him. If Alex ate the minimum number of cakes possible (1), and Danny ate a minimum of 1, that leaves 9 cakes left for Charles and Ben. If Ben had eaten 5 or more cakes, he would know Charles couldn’t have possibly ate more than him. Ben therefore ate less than 5 cakes and more than 1. Ben’s possibilities: 2,3,4

Charles knew all the above but he still wasn’t sure if he had eaten more than Ben or not. If he ate 2 or less, he would’ve answered “no” to Ben’s question (since there is a possibility that Ben has eaten 2, and if that’s the case, Charles would answer “no” even if he ate 2.). If he ate 5 or more, he would’ve answered “yes”. Charles’ possibilities: 3,4.

Then Danny (and only Danny!) figured out the amount of cakes each person had with only the added knowledge of his own. This means there is only one combination of the possible amounts of Alex, Ben, and Charles that can yield the amount Danny ate.

The only way to get a sum unique to one combination are if each person ate his maximum or minimum amount. The maximum sum of all the three is 4+4+4=12. That can’t be right since there’s only 11 cakes! The minimum is 1+2+3=6. This leaves 5 cakes for Danny.

Since Danny knew he had 5 cakes and that the only way for him to have 5 cakes is if Alex had 1, Ben had 2, and Charles had 3, he was able to figure out the amount each person ate.

So Danny had 5 cakes, Alex 1, Ben 2, Charles 3.

I could have given more details like using symbolic logic but that would just complicate an otherwise simple problem. So, I hope that this explanation is detailed enough. Feel free to ask me questions if you find anything confusing.

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19. Jeff Rab says:

Love the puzzle! But I came up with 2, 2, 2 and 5. I based that answer on they didn’t know how many “cakes” they ate and Alex specifically asked Ben if he ate more “cakes” than him. The plural of “cakes” indicates more than one cake….at least to my pea brain…thus the first three ate 2 cakes each and the last ate 5.

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• Oh I also want to add that it’s not possible for Charles to have eaten 2 cakes since he would have answered no. Note that Charles was already aware that Ben ate at least 2 cakes. This means that if Charles ate two cakes, he would know that he didn’t eat more than Ben so he’d answer “no”.

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• Jeff Rab says:

Thank your for the very detailed explanation! I’m going to go off in a corner, pull this puzzle out again, and sit and scratch a little while to figure it out!

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• You’re welcome 🙂

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