# Confusion on a math puzzle

One of my colleagues asked me below puzzle,

Three friends went to a picnic. 1st and 2nd friends collected woods for cooking where 3rd friend did not participated because he was sick. 1st friend collected 3 woods and 2nd friend collected 5 woods. Now 3rd friend (sick one) have 8 dollars with him. As he could not able to participate in wood collecting he wants to give this 8 dollars to other two friends as gift. Now, what should be the proportion of distribution based on other friends contribution?

Note: Contributions are calculated on collected woods only.

1st friend gets 2, 2nd friend gets 6,

As below calculations,

1. Total cost = 8 unit
2. Par friend cost = 8/3 = 2.6666666 (as 3rd friend participated in eating)
3. Countable contribution (8 - 2.6666666) = 5.333333333 (excluding self)
4. So, 1st friend gets (3/8)*5.3333333 = 2
5. So, 2nd friend gets (5/8)*5.3333333 = 6

But, my colleagues denied and claimed that correct answer is,

1st friend gets 1, 2nd friend gets 7

As below calculations,

1. Total wood = 8 unit
2. Par friend cost = 8/3 = 2.6666666 (as 3rd friend participated in eating)
3. So, 1st friend gets (3 - 8/3) = 1/3 = (0.125 of 8)
4. So, 2nd friend gets (5 - 8/3) = 7/3 = (0.875 of 8)

Thus,

1st friend gets 1 (12.5% of 8), 2nd friend gets 7 (87.5% of 8)

### Update

I became aware from Lanny Stracks comment that I have done a wrong calculation in my step 5. Thanks Lanny Strack.

• If the 8 dollars was for the wood collecting, where does the food come into it? There is no mention of food in the question, other than the fact that it is a picnic. What is the value of food compared to the value of collecting wood? May 3 '20 at 20:12
• Your math is wrong in step 5 of the first answer. May 3 '20 at 21:51
• Ohhhoo. Thats really a very silly mistake. Thanks a lot @LannyStrack. However, I got the concept from greenturtle3141's answer. I have appended an update. May 3 '20 at 22:23
• This is a very old puzzle, by the way. Here it is in Henry Dudeney’s 1907 book The Canterbury Puzzles: gutenberg.org/files/27635/27635-h/27635-h.htm#p31 May 4 '20 at 0:51
• @RomanOdaisky thats great! thanks May 4 '20 at 0:58

I'm unsure why you calculate a "countable contribution". The key idea is that each friend has, initially, a work obligation of 1/3 of the work. The sick friend was unable to contribute as such. Therefore, their work was passed onto the other two friends. The sick friend is paying their friends according to how much work of the sick friend's workload they did.

That's why you ignore 8/3 of the wood from each of the two friends that did anything; that share of wood collecting was an obligation. The extra wood they collect is what you pay them for, because that was wood that you were supposed to collect.

For a simpler example, let's say me and my two best friends are making pizzas. We plan to make three pizzas in total. So logically, we each make one pizza. However, I am not feeling well, so I can't do my share of the work. So, friend A makes an extra pizza. In the end, A made 2 pizzas and B made 1 pizza. If I have 3 bucks, how do I distribute the money?

B didn't do any extra work, so I have no reason to pay her. Because of my laziness, all the work that I was supposed to do was taken care of by A. So it is only logical that I pay A all of my 3 bucks. I mean, she was the one that made my pizza, it makes sense!

• Excellent! your simple example helped me to understand my problem. Thanks a lot. May 3 '20 at 23:43

I'm going to answer a different question here.

There were no "8 woods". The guys brought 3 and 5 loaves of bread instead. This way we don't need to care about whether the money guy wanted to pay "per wood-gathering hour" or "per woods".

The "8 woods" didn't unexplainably transmogrify into "8 foods" halfway through. Instead they were loaves of bread all along, and each person ate an equal portion.

Everyone ate $$\frac{8}{3}$$ loaves of bread. The 3-bread guy sold $$3 - \frac{8}{3} = \frac{1}{3}$$ loaves of bread, because that's all he had to sell after eating his part. The 5-bread guy sold $$5 - \frac{8}{3} = \frac{7}{3}$$ loaves of bread, which is 7 times as much, and so is entitled to seven times the payment.

The original question, as written, is poorly defined, and allows for all kinds of misunderstandings. Maybe tell your friends to be more precise when posing classical math puzzles? :-)

The reason you got a different answer is that you invented a "Countable contribution" value, which very surprisingly gave neat integer solutions adding up to the correct amount, when multiplied by the fractions. If you figure out what that value actually means, and what would be the appropriate uses of such a value, I'm sure you'll find that those calculations share no other merit than arriving at a plausible sounding (yet mistaken) result.

Let's assume that each wood comes in 3 equal pieces. This means that the first friend brings 15 pieces and the second one brings 9. Since each of the 3 friends are supposed to bring 8 pieces each, the first one brings 7 pieces more, and the second one brings 1 piece more. Which means that the first friend gets 7 dollars and the second one gets 1.

• Yes, that can be another visualization of the answer. Thanks. May 4 '20 at 10:47

There's a lot of extra stuff being added to the question beyond the original - talk of "obligation" and "food" just distracts from the structure of the problem.

The "trick" here is to remember not only what the first two friends brought to the table, but also to account for what they have consumed. Friend 3 isn't paying for everything, just for his share.

Friend 1 contributed $$3/8$$ and consumed $$1/3$$, for a surplus of $$1/24$$. Friend 2 contributed $$5/8$$ and consumed $$1/3$$, for a surplus of $$7/24$$. Friend 3 contributed $$0/8$$ and consumed $$1/3$$, thereby consuming the surplus $$8/24$$.

The surplus numbers cancel out, and that's what Friend 3 wishes to pay for. Friend 2 contributed $$7$$ times as much of what Friend 3 contributed, so that friend gets $$7$$ times as much. Conveniently, that amounts to 7 dollars.

8 Dollar, 8 wood.

Cost of wood per dollar = (8 wood)/(8 dollar) 1 wood/dollar

first friend

You buy 3 wood from your first friend: 1 wood/dollar * 3 dollar = 3 wood -> 3 dollar

second friend

You buy 5 wood from your second friend: 1 wood/dollar * 5 dollar = 5 wood -> 5 dollar

• Not really. 3rd friend wanted to gift the 8 dollars according to others contributions. All friends ate same amount of food right? May 3 '20 at 19:51
• But assuming that they each bought 1/3 of the food each, it should be an equal amount of food bought + eaten -> only owes for wood gathered? May 3 '20 at 19:55
• Actually food was cooked with collected woods. No food were bought before. May 3 '20 at 19:57

Let's say to eat you have to pay 8/3 wood to eat dinner. A collected 9/3 wood and B collected 15/3 wood. C doesn't have any wood, so he has to buy wood from his friends. A and B both have to keep enough wood for themselves to pay for dinner, so C has to buy 1/3 wood from A and 7/3 wood from B so they can all have enough wood to pay for dinner.

Look at it as if you guys needed 8 wood to cook your food. To be fair, each of you should contribute 8/3 wood. Since C is sick, the others collect extra wood to pick up the slack so they have enough for cooking.

1st friend collected 3 wood. He should have collected 8/3 wood for his fair share anyway, so since he collected 9/3 wood, he contributed 1/3 extra wood to the total because 3rd friend was sick.

2nd friend collected 5 wood. He should have collected 8/3 wood for his fair share anyway, so since he collected 15/3 wood, he contributed 7/3 extra wood to the total because 3rd friend was sick.

Since 3rd friend was sick, 1st and 2nd friends collected a total of 8/3 extra wood over and above what their fair share would have been anyway, so 3rd friend should divide his \$8 among them based on how much they collected in excess of their fair share

There are total 8 woods, so each of them will get 8/3 wood .

1st friend collects 3 woods out of 8 woods collected in total,uses 8/3 wood ,leaving 3- 8/3= 1/3 wood for 3rd friend.

2nd friend collects 5 woods out of 8 collected in total, consumes 8/3 wood, leaving 5-8/3= 7/3 wood for 3rd friend.

So net contribution of 1st friend is 1/3, and second friend is 7/3.( 1/3+7/3 = 8/3 for sick friend).

So, sick friend will distribute his 8 dollars in the ratio of 1/3 : 7/3, which comes out to be 1 dollar for 1st friend, 7 dollars for second friend.