This puzzle is dedicated to our most recent 10-puzzle-points winners: Donna, Dennis, and Katrina.

Katrina and Dennis, who happen to be a husband and wife, decided to lease the popular spot no longer occupied by the Hollywood video and open a restaurant. But what is a restaurant without a top chef? They lured in a rising Food Channel star - chef Donna. Smart and creative as usual, they decided on a restaurant gimmick - college students employees will help your kids do homework while you can enjoy the food, wine and an uninterrupted conversation.

The DDK Family Restaurant has been open for only a year and already got the prestigious Best Afterschool and Less is More awards. The local news channel did a poll of the restaurant visitors and found that 70% had their own single favorite among the 10 amazing dishes Donna created and always came back for it. The remaining 30% percent wished there would be some new entries every time they come - something that Dennis and Katrina didn't offer so far. Dennis and Katrina invited Donna for a brainstorming session to interpret the results of this poll and see how they can make their customers even happier. Donna said that she can not offer more than 10 choices each day. Can you help them figure out how frequently and how many items on the menu they should vary?

Answers accepted all day long on Friday, on our Family Puzzle Marathon. They will be hidden until Saturday morning (EST) and everyone who contributed something reasonable will get a puzzle point. If you feel some information is missing - make any assumptions you want, just state them specifically.

## 11 comments:

(Mostly non-math answers I suppose.)

Soups? Desserts?

Sounds like business is very good, and if ain't broke....don't mess with it.

The goal is to "make their customers even happier?" Not exactly. The goal is to serve more meals by gaining more traffic. ANd not lose any regular customers if possible. And chefs and owners wrestle with this all the time, everywhere.

If 70% of the customers return for their individual favorite dishes, we'd better not tamper with those dishes, not much. The local news channel's survey may not have determined WHICH dishes were the most popular, but the owners most certainly do know; and the owners could also tell us which items on the menu are "less" popular. Those less popular items are the ones to offer less often.

One solution is to introduce a standard weekend menu. Another is to have a "nightly special," and these could be surprises, or they could be the Tuesday special, the Wednesday, etc.

I certainly wouldn't claim to be a math whiz and this is my first time to write so please go gently on me ;c).

I would think the owners should poll their 70% satisfied customers to find out what their favourite dishes were. (I worked with 100 total customers surveyed to give me some easier figuring.) This would mean my satisfied customers equal 70.

I can only safely assume that there is an equal proportion of favourite dishes chosen - 7 people (7%) picked each of the ten meals as their favourite. In order to satisfy the 30% who wanted new ENTRIES (emphasis on the plural - assuming the change of one meal wouldn't be enough), I would have to change 2 meals on my menu each day. That would mean that I would possibly leave 14 of the 70 satisfied unhappy when their favourite meal disappears from the menu.

However, if I add my new percentages - 30% who are now satisfied because of variety and the remaining 56% who are still satisfied because their favourites are still offered I would up my customer satisfaction to 86%.

If I went further and started reintroducing one of the original 10 meals that I had dropped during this change (without dropping one of the 8 that I kept) I could expect to gain back another 7% on alternating days thus eventually bringing up my satisfied customers percentage to 93%. (This is of course assuming that the 30% who wanted variety will still see this as variety as the menu will be constantly changing).

Does that make sense or is my thinking flawed??

Donna wants at least 70% to have their own single favorite among the 10 dishes. Also, 70% of her customers favor her.

The probability that Donna will receive at least 70% of people votes would be:

P(Y/n>or= 0.70) and using the normal approximation to the binomial distribution with p=0.7 and p being the probability that a random person in her restaurant favors her 10 dishes choice with no change.

I found P(Z>or =0=1/2)

she has 50% chance to make people happier.

I will not vary the menu for now. If she does so, she needs more information on people preferences, range of prices that people are ready to pay...

Maria, I received the surprise and I like it!

Thanks!

These were my thoughts about the menu composition:

The intuitive answer seems to be 7 to 3: leave 7 items on the menu and offer 3 new items every day, to satisfy the 70%-30% patron preferences ratio.

But then the chances that any random patron will be happy are: 0.7 x 0.7 + 0.3

that comes from (probability that he is among 70% who loves ordering the same stuff) x (probability that the item that he likes is still on the menu) + (probability that he is among those 30% who like variety) x 1

0.7 x 0.7 + 0.3 = 0.49 + 0.3 = 0.79 or 79%

Can we make it higher?

What if Dennis, Katrina and Donna will offer only one new item every day with 9 items being constant.

In that case the probability of any random patron being satisfied is: 0.7 x 0.9 + 0.3 = 0.63 + 0.3 = 0.93 or 93%

This seems to be as high as we can go. Keeping all 10 items the same will leave only 70% of the people happy.

Of course here I make an assumption that one new item is enough to satisfy novelty-seekers. Beth seem to come to a very similar solution and answer but she pointed that I used plural "items" instead of a singular "item" when describing preferences of the 30% that are seeking some novelty in the menu. So, her answer was 2 new items every day.

All Tom's suggestions are very interesting. Anne-marie seems to be a queen of the binomial distribution. This is at least a second time she is using it in a puzzle solution. I am not sure I completely understand her answer as I am quite rusty in this but I am sure she is at least partially if not absolutely right. Perhaps Kim can comment on this.

A puzzle point for everyone who played - Tom, Beth and anne-marie. I am surprised we didn't hear from the restaurant owners. D,D&K - what do you think?

I think I was under the weather, and Katrina was out of town... :) But, when I looked at it, I thought it was less a math problem than a customer service problem. What I saw was that the survey did not ask people WHAT dishes they liked, as Tom said. As an extreme case, all 70% with a favorite could like the same dish, and I'm in trouble if I remove that one. The solution I had was to see what was selling and what wasn't, and replace the poor performers, one at a time, and see what the impact was on my sales. Unfortunately, I went to bed and slept, and missed the deadline.

I do, however, like both 93% solutions, if the assumptions hold! Thanks for featuring us!!

Sorry, Marie. I read this at work yesterday, and began making assumptions, and writing down a rotation of dishes....and then forgot about it after I got home and had appointments and family stuff going on. I really wanted to participate. My husband (and kids)would definitely get a kick out of me being named "chef"!

Oops....I called you Marie, I think....Maria. Sorry. :)

Just have to say things in a confident way to make people believe it!

I love math except probability and proof however, I am taking right now probabiliies/ statistics classes so I try to use it, it helps me to understand but it does not mean the answer is right!

Just a student who try to apply what she is learning in math.:)

For the restaurant, I think the owners could invite you!

Another non-math answer. 70% are happy, so don't mess with success... too much. 30% want something different. Daily changes are difficult / impossible. What about weekly?

Change 2-3 items per week. Next week, change them BACK and keep the most popular of the 2-3 as you change 1-2 different items for the second week. And so on.

So... week 1, items 1-10 as is. Week 2, items 1-8 (a conservative approach, since we don't know which of the items is most popular) plus add items 11 & 12. The following week, if one of 11 & 12 is as popular as the other items, keep it. Put 10 back on the menu, changing out item 8 for item 13. Week four, put item 9 back and switch out 7 for the most popular of 13 and (the most popular of 11 & 12).

And so on.

The chef is happy, she's only making 10 items per week. The customers are happy, because along with a little variation, there are the old standard menu items that they love. And the kids are happy, because they get to keep coming and getting help with their homework!

I wish I lived closer!!

Hire that guy on "Kitchen Nightmares".

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