# How should they divide the money?

Three friends go on a picnic.

John brings a loaf of sliced bread. Raj brings sliced tomatoes and Von brings a package of sliced cheese.

They make sandwiches and eat them. John’s sandwich has 2 slices of bread, 2 slices of Tomatoes and one slice of cheese. Raj makes a sandwich with 2 slices of bread, 4 slices of tomatoes and 3 slices of cheese. Von’s sandwich contains 2 slices of bread, 2 slices of tomatoes and 4 slices of cheese.

Now an old acquaintance Sergio drops by and is hungry. They offer him their food.

He makes 2 sandwiches using 4 slices of bread and 1 slice of tomato, no cheese.

He is happy and before he leaves he gives them 6 pesos as a goodwill gesture.

“How should we divide this?” asks John.

“Well he only ate bread and tomato so John and I should get it” declared Raj.

“That is not fair. You guys ate lot of my cheese” complained Von.

They could not resolve the division of money issue so they found a mathematician friend and asked him to decide.

“How much did it cost you to get the food? Give me details” He said.

John: "My loaf of bread had 12 slices and cost me 12 pesos"

Raj: "I bought 2 tomatoes for 10 pesos each and cut out 10 big slices"

Von: "I bought a packet of cheese slices for 20 pesos and had 10 slices"

Then they also told the mathematician how much they all ate.

The mathematician rendered his verdict: “You all should share the 6 pesos equally”

What logic did he use? Do you agree?

Basically, given the prices of each piece (1 peso per slice of bread, 2 pesos per tomato slice, and 2 pesos per cheese slice), after John, Raj, and Von eat their pieces, John owes Raj 2 pesos, Raj owes Von 2 pesos, and John and Von are even.

Given what Sergio eats, he owed John 4 pesos and Raj 2 pesos. Suppose he distributes his 6 as such. Then, after John pays Raj what he owed, and Raj pays Von what he owes, they're each left with 2 pesos out of Sergio's original 6.

The logic the mathematician used was simply that if someone consumes something brought by somebody else, they should pay that person exactly what that person paid for that piece at the shop. (e.g. I should pay Raj 2 pesos for each tomato slice I eat.) Do I agree with that logic? Well, that's a normative question, not really suitable for an answer to a riddle. I personally wouldn't be so stringent, but I guess that's unimportant.

• The problem I see with your solution is what happens if Sergio overpays or underpays? If he gives 12 pesos instead of 6, according to your logic he’d give 8 to John and 4 to Raj. After John pays Raj, and Raj pays Von they’re left with 6, 4, 2. But if Sergio gives 3 pesos instead they’re going to be left with 0 for John, 1 for Raj and 2 for Von. Doesn’t seem fair for the amount change like that. Commented Feb 24, 2020 at 17:18
• @Amorydai My logic only covers what each person ought to pay, not what to do with extra payment or unaccounted for debt. Before Sergio enters the picture, John should be at -2, Raj at 0, and Von at +2. Suppose Sergio doesn't pay at all. What would be "fair", then? Also, how do you get that my logic implies that if Sergio pays 12, John gets 8 and Raj gets 4? I don't see how that can be concluded from what I said. My logic would only dictate John gets 4, Raj gets 2, and the remaining 6 is basically charity unaccounted for. Commented Feb 24, 2020 at 17:53
• I see. Well, to me, the most important question (and the one specifically mentioned in the problem) is “since Sergio didn’t eat any cheese, should Von get anything at all?” From my understanding of your answer, where Sergio only pays John and Raj it would seem the answer is no, Von doesn’t get anything. So if Sergio did pay more than he should have, John and Raj would benefit unduly at the expense of Von. Of course you don’t specifically address that situation, so the point is moot - and vague. I guess it’s just lucky Sergio paid exactly what he did. Commented Feb 25, 2020 at 0:29

Firstly, let's assume that the uneaten food was not brought by the friends at all, because it doesn't play any role since it all remains with its original owners. So, we can assume that only 10 slices of bread (worth 1 peso each), 9 tomato slices and 8 cheese slices (both worth 2 pesos each) exist.
Now, let's count the balance of each friend:
John: brought 10 slices of bread which cost 10 pesos, but his sandwich costs only 8 pesos. So, his balance is +2 (that means that somebody, we don't care who exactly, owes him 2 pesos).
Raj: brought 9 tomato slices costing 18 pesos, value of his sandwich is 16 pesos. Balance is +2.
Von: brought 8 cheese slices costing 16 pesos, value of his sandwich is 14 pesos. Balance is again +2.
Sergio: brought nothing (0 pesos), his sandwiches cost 6 pesos. Balance is -6 (as expected, the sum of all balances is zero).
Now it's obvious that it's fair to divide 6 pesos equally between the 3 friends (2 pesos each).

The mathematician is right and I agree with his decision. When matters of dividing resources comes up the best way to find a solution is just think of capitalism. You can think of the picnic as a “bank” of sorts. John brings 12 pesos worth of bread and “sells” his bread to the bank. Raj and Von each sell 20 pesos worth of food to the bank. Now John can buy 12 pesos worth of food from the bank in any proportion he wants, and Raj and Von can each buy 20 pesos worth of food.

The problem didn’t say specifically what happens to the left overs, but I will assume John gets all the leftover bread, Raj the leftover tomatoes and Von the leftover cheese. So with the sandwiches they made and the leftovers they got, John ends up with 4 pieces of bread, 2 tomato and 1 cheese - that’s 10 pesos so he should have 2 pesos in his pocket. Raj got 2 bread, 5 tomato and 3 cheese - 18 pesos worth of food so he should have 2 pesos in his pocket. Von got 2 bread, 2 tomato and 6 cheese - again 18 pesos worth of food so he should have 2 left.

You can think of Sergio’s pesos as him buying 6 pesos worth of food from the bank and leaving his special “Sergio’s” pesos in the bank. Each of the friends can then buy 2 of Sergio’s pesos from the bank and none of them can buy more than 2 pesos because that’s all each of them has left.

That mathematician was a smart guy!

• What's the point of buying pesos for pesos? To me it seems meaningless. Also it's not clear why any of the participants could not have more money than those obtained from the bank, the problem did not say they do not have (other) money. Commented Feb 25, 2020 at 9:32
• @Andrew Savinykh the point is to keep track of everybody’s finances. Also the pesos are virtual, when John “sells” the bread to the “bank” nobody is giving him any pesos. So when Sergio left 6 real pesos in the bank, John can use his 2 virtual pesos and exchange them for the real pesos in the bank. It doesn’t matter to this problem if they have extra money because we’re determining the fair distribution of Sergio’s pesos and the food. So for the purposes of this problem only virtual pesos can be used to buy stuff out of the bank. Commented Feb 26, 2020 at 22:05

This is not merely a Mathematical problem.

but an Accounting One

The logic goes like,

Step 1

Capture all transactions related to initial pool of resources
Account      Dr   Cr
John            0      12  -> value in Pesos for the bread that he pitched in
Raj               0     20
Tomato      20        0
Von              0     20
Cheese      20       0

Step 2

Capture transaction related to consumption
Account      Dr   Cr
John            8       0
Tomato        0       4
Cheese        0       2

Raj              16     0
Tomato        0       8
Cheese       0       6

Von              14     0
Tomato        0       4
Cheese        0       8

Step 3

Capture the transaction with Sergio (both as Sandwich & Cash)
Account      Dr   Cr
Sergio         6       0
Tomato        0       2

Sergio        0       6
Cash          6       0

Step 4

Now observe the closing value of each Account
John - 4 Cr
Raj - 4 Cr
Von - 6 Cr
Tomato - 2 Dr
Cheese - 4 Dr
Cash - 6 Dr
Sergio - 0

Step 5

Since each of John, Raj & Von will take back the remaining of whatever they've brought, let's account that into the closing balance as well
John - 2 Cr (4 Cr against 2 Dr of Bread)
Raj - 2 Cr (4 Cr against 2 Dr of Tomato)
Von - 2 Cr (6 Cr against 2 Dr of Cheese)

Now it's clear that each of them have net claim of 2 Pesos each, which was rightly paid by Sergio

• Wow @Kalaivanan. You must be expert in numbers!
– DrD
Commented Feb 25, 2020 at 13:16

Being a true mathematician, the friend has little interest in mindless summation of numbers.

However,

as his specialisation is type theory, he notices the obvious type mismatch between money spent and food consumed. Since there's no clear bijection (only the initial one-way conversion), food and money must be divided equally in separate instances, if some equal division even exists.

From that,

it's trivial to do the money side of the division. 6 pesos can be divided by three people. The food side, on the other hand, seems much more tricky since they even have different preferences in what they consume. But they didn't ask about how to balance out the leftovers, so the mathematician goes happily back to his study.

• Liquidizing a single slice of tomato seems tricky to an economist. The same view is not shared by a chef with a blender. Commented Feb 25, 2020 at 0:00
• They are friends right? So just share and be happy :) Commented Feb 25, 2020 at 15:41
• This is truly the best answer. Commented Feb 25, 2020 at 17:19

Late to the party, but still want to answer.

John looked at how much they each spent (John: 12 pesos, Raj: 20 pesos, Von: 20 pesos), the value of what they consumed (John: 8 pesos, Raj: 16 pesos, Von: 14 pesos) and the value of their respective remaining food (John: 2 pesos, Raj: 2 pesos, Von: 4 pesos). If we look at the amount spent, less the amount consumed, plus the amount of food left over that each of them purchased, then everyone is out an equal amount of money-- 2 pesos. As such, it makes sense to split the six pesos evenly, so that everyone has either eaten or retained as much food as they purchased.

Fun riddle!

Being a theoretical mathematician myself, I don't agree. The accounting solution offered in other answers is a: doubtful, esp the slice of tomato cannot be taken home at full prize. b: not a mathematical solution i.m.h.o. An alternative solution would be to split equally as a 'social' solution. It is simple, and we being among friends that seems fair. Again however, that would not be my ruling as a mathematician. I would look at the actual picnic. In total 44 pesos of food has been eaten; So Johns contribution is 10/44, Rays 18/44 and Vons 16/44. John gets 6*10/44 pesos , Ray gets 6*18/44 pesos, Von gets 6*16/44 pesos. The handling of different contributions and different consumption are separate problems, which I am willing to rule over for 6 pesos. The fact that there is no such thing as a 1/44 pesos: as a mathematician I don't care.