Upon the advice of my friend Kate Litvak, I took a short summer vacation to Walt Disney World.Â (In reality, the trip was a work excursion, to meet up with the other Professor Nowicki to work on an executive compensation paper.Â But the good news is that the other Professor Nowicki was then at Walt Disney World, so I was able to treat the work trip as a bit of a vacation.)Â

The following vexingÂ math problem arose on the trip, and I am offering it to the group for your consideration:

Dad Nowicki,Â ProfessorÂ Nowicki, and the Other Professor Nowicki go to lunch.Â Â The Other ProfessorÂ Nowicki, due to a professional relationship with “The Institution,” gets a 20% discount for all on lunch.Â The 20% comes off of the total bill, before tax.Â Gratuity of 18% is automatically added before the discount is taken.Â Tax in this state is 8%.Â The total bill comes to $44.21.

How much was the original price of each person’s meal?

I am not going to tell you how long it took me to calculate the answer to this question – I blame it on the post-lunch food coma.Â Â Tell me how long it takes you.Â The shortest time *wins* the lingering Univ. of Richmond.Â John Armstrong and James Grimmelman are disqualified from the competition for obvious reasons.

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Anthony,

Thanks for pointing out that Error. I am ashamed.

I got $14.45 per person, took about 2 minutes. I treated it as a series of seperate problems.

1. What was the bill without the included tax 44.21/1.08= 40.94

2. What was the bill without a 20% discount? 40.94/.8= $51.17

3. What was the bill without the gratuity added on? 51.17/1.18= $43.36

4. What was the price per person? 43.36/3=$14.45

I wasn’t satisfied with that answer though.

This price seems a little strange for a buffet unless drinks were a seperate price and each person had a drink with a slightly different price. The resulting number would then include the average drink price. If two people had a large drink at $2.19 a piece and the 3rd had water, Each meal would have cost $12.99.

$12.99 per person seems more reasonable for a buffet lunch in Disneyworld.

Antony Page:

I donâ€™t see how your formula takes into account duplicate combinations.That’s because I treated the identities of the three eaters as significant. I assumed-slash-stipulated that we would count the situation in which Dad pays $15.40 and Professor pays $16.10 as different from the situation in which Dad pays $16.10 and Professor pays $15.40. If you decide that only the prices, rather than who paid what, are significant, than yes, different answer.

Calling each solution a “combination,” as Kate did, might suggest that order is irrelevant, because one would say “permutation” if order is significant. I’d argue, however, that this isn’t a problem of combinations and permutations at all: we’re actually counting “partitions.”

Oh, câ€™mon. If I wrote â€œElizabeth ate a fillet mignon,â€? wouldnâ€™t your first reaction be â€œaha! We canâ€™t just divide the total bill into three equal parts!â€?? So, why is the substitution of a cow for a dog made it so confusing? Are you saying it never occurred to you that the best Chihuahua is grilled Chihuahua?

I thought we already had everything set up for the holidays, no? You babysit our kid; we are out skiing. We already ordered a full set of Saved by the Bell DVDs, as you asked. Has anything changed?

Thanks for the welcome – with a blog thread this wonderful, I felt that I could lurk no more.

BTW, did your server gets 18% of the buffet price pre or post discount? Inquiring minds want to know, as this is one difference between Rob’s and Urijah’s answers. But either way, neither price seems Disneyesque. Perhaps pricing wasn’t considered (an omission clearly “not in good faith”). One also wonders how much gratuity would be added if you’d actually got waiter service …

I love you all. This is a wonderful thread. Perhaps the finest thread ever.

Just so that we are clear, this collective body has determined that the price per person of the Fort Wilderness buffet is between roughly $14 and $9 million. Good for us.

Let me say two things: 1. Each person’s buffet price was the same, and the 20% off applied to each person’s meal. 2. I am pretty sure I have the total lunch price wrong. I’ll have to ask my sister to check her Mastercard statement. I will report back.

God bless you all – I have not laughed this hard in a while. And, Kate, I still do not see the relevance of your Chihuahua comment. To be honest, when I first read it, I just thought it had something to do with the language barrier. While we are on that topic, will I be invited to your home to celebrate the holidays this year? Just askin’.

P.S. Antony Page, good to have you here.

Rob: 42.35/3 doesn’t equal 14.78.

James: my math “skills” are archaic, but I don’t see how your formula takes into account duplicate combinations. In other words, if the three meals are priced at x, y, z, we have only one combination of prices regardless of who consumes which meal. (i.e. Dad Nowicki eating the meal priced at x in x,y,z; the y in y,x,z; or the z in z,x,y; are all only one combination of prices.)

I get 9,823,528.

There’s a total of $44.34 to be divided among three meal prices. That’s 4434 cents to be partitioned among three labelled meals. The condition that each meal costs at least one cent means that this problem is equivalent to partitioning out 4431 cents among three labelled, possibly empty, meals. This problem, in turn, is equivalent to picking (in order) two integers whose sum is less than or equal to 4431 (the price of the third meal being whatever is left). We can write this expression as:

\Sum_{i=0}^{4431} \Sum_{j=0}^{4431-i} 1.

The outer sum corresponds to picking the price of Dad Nowicki’s meal. The inner sum corresponds to picking the price of Professor Nowicki’s meal, after taking out the price of Dad Nowicki’s meal. The 1 corresponds to the single unique price left for Other Professor Nowicki’s meal once the prices for the first two meals have been set. This simplifies to

\Sum_{i=0}^{4431} (4431 – i + 1)

because the inner sum is equal to the number of terms in the sum. (n times 1 is n.) We can now add the 1 to the 4431 to get 4432:

\Sum_{i=0}^{4431} (4432 – i)

Splitting the sum over the two inner components gives:

(\Sum_{i=0}^{4431} 4432) – (\Sum_{i=0}^{4432} i)

The first term here is easy; it’s just 4432 added up 4432 times (since the index i starts at 0):

(4432 * 4432) – (\Sum_{i=0}^4432 i)

The second term is only slightly more complicated. It’s a triangular number. Using the formula for the sum of the first n integers — (n * (n-1)/2) — we can rewrite our equation as:

4432^2 – ((4432)(4431)/2)

Doing out the multiplications, we get:

19,642,624 – (19,638,192/2)

Performing the division by 2 gives:

19,642,624 – 9,819,096

Which yields a final answer of:

9,823,528

Arghhh, and nobody picked up the indeterminacy point from my Chihuahua comment! My point was that even if know exactly what Elizabeth ate, we still canâ€™t tell how much each personâ€™s meal was.

Next question: contrary to Jamesâ€™ comment above, the number of possible solutions isnâ€™t quite infinite, so long as meal prices involve only whole cents. Hence the question: how many combinations of meal prices are possible, assuming that no meal costs less than 1 cent, each meal costs something, and prices involve only whole cents?

Thanks, James. I also realize I didn’t post my time. It took my about 8-10 minutes because I did it by hand and make the mistake of thinking .08(.8)=.64 instead of .064 so I had to recalculate.

The answer is indeterminate; there are infinitely many possible solutions. The comments above assume that the price of each person’s individual meal was the same, but that information wasn’t stated in the problem. Given that additional assumption, Rob is correct.

Among one group of my friends, the rule was that the “youngest non math major” present had to figure out how much each person should pay when check-splitting time came.

I got a different solution.

I figured there are three “terms” that add up to the toal of 44.21.

The first term is the “Price minus Discount” which is .80P. (P-.20P).

The second term is the “Gratuity” which is .18P. (18% of the original price).

The final term is the tax which is .08(.80P). (Eight percent of the discounted price).

Summing these terms yield this equation:

.80P + .18P + (.08 * .8P) = 44.21.

I solved for P and got 44.21/1.044=42.35

Dividing that by 3 I got 14.78.

Elizabeth Nowicki ate a live Chihuahua. You heard it here first.

14.4545, about a minute. (I cheated though…in Mathematica

Solve[(x + .18 x) – .2 (x + .18 x ) + .08 ((x + .18 x) – .2 (x + .18 x )) == 44.21, x]

{{x\[Rule]43.3635}}

(x/.%)/3

{14.4545}

WB – per person. The bill was $44.21 for three (we’re talking about Walt Disney World, remember!)

$43.36. 4.5 minutes.