## Good Students

We’ve been working hard on getting the Smartkit Forums ready. Come Monday, we’ll officially open them up. You’ll be able to:

- post some of your own puzzles
- play forum games
- pick the brains of other smartkit members

OK, here’s Thursday’s challenge. Can you figure it out?

Three students, each carrying an equal number of apples, were met by 9 teachers, who asked for some of them. Each student having given to each teacher the same number, it was then found they had all equal shares. How many had the students at first?

Will unmask submitted answers Monday morning, right before we make the forums live.

marioberges| Profile May 8th, 2008 - 8:19 pmEach student started with 12*n apples, where n is an integer > 1.

Then each student gave each teacher n apples.

Since three students gave each teacher n apples, each professor in the end had: 3*n apples.

Each student, after giving apples to 9 teachers was left with (12 – 9) * n = 3*n apples.

The number of apples each student/teacher had in the end is the same.

runninfool| Profile May 8th, 2008 - 9:09 pmEach student starts with 36 apples. They give 3 apples each to each teacher. Every teacher and all three students will end up with 9 apples.

falwan| Profile May 8th, 2008 - 10:21 pmEach student had (12n) apples…n=1,2,3…

each student gave each teacher 1n apple each…

12n – 9n = 3n apples left with each student…

1n + 1n + 1n = 3n apples left with each teacher…

Everybody will be left with (3n) apples…

Hope this one has no twist…

Congrats.! on your forum project!

zxo| Profile May 8th, 2008 - 10:42 pmThe hard part was interpreting the second sentence.

Each student had 12 apples. They gave 9 away to the teachers, leaving them with 3. The teachers received one apple from each student, so they also had three.

This also works for multiples of 12, 9, and 3 — for example the students may have had 24 to start, gave 2 apples to each teacher to leave each with 6.

joe| Profile May 9th, 2008 - 6:36 amIn the end the 3 students and 9 teachers need to have the same number of apples. So the total number of apples needs to be divisible by 3 (to start with) and also 12 (total number of poeple). Starting with a total of 36 (12×3) apples, each student would start with 12, they would each give 1 apple to each teacher and be left with 3 each. The teachers each receive 1 from each student so they get 3 aswell. So we can confirm that each student starts with 12 apples each (or a total of 36 between them).

tumeke| Profile May 9th, 2008 - 7:06 amMultiples of 12.

MattHogan| Profile May 9th, 2008 - 9:05 amMy first guess would be 12*X, where X is the number of apples given to the teachers. If the teachers are given only one apple from each student, they’d all end up with 3 apples, and if the teachers were given two, they’d all end up with 6 apples.

I could be wrong though, but I’m pretty sure that works.

lograh| Profile May 9th, 2008 - 9:43 amSo the number of total apples doesn’t change, and is evenly divisible by both 3 (since they each were carrying an equal number at the start) and 12 (since they all had equal shares at the end). Since 3 is a divisor of 12, we can just focus on the multiples of 12.

Thus potential total numbers of apples are of the form 12n, for n=1,2,3,… Here it is helpful to do a bit of sanity checking, though. If there were only 12 apples to start, then each teacher would end up with 1 apple total, but since each teacher got an equal number of apples from each student, that would imply they got out their knives and each student gave each teacher 1/3 an apple. I’m going to guess they were not quite so crafty and this puzzle likely stays within the realm of the integers, so 12 total apples is out. Similarly, 24 48, 60, and so on all have the teachers ending up with a number of apples that is not evenly divisible by three (the number of students who each gave each teacher an equal number of apples). So we are left with possible total number of apples of the form x=12n where n=3,6,9,…

Thus possible individual number of Apples “carried” by each student are of the form 12n where n=1,2,3,… (x/3 where x is from above and we just divide out the three from the iteration factor)

So there could have been 36 total apples, with each student carrying 12 to start, and everyone having 3 at the end (each student giving one apple to each teacher).

Or there could have been 72 total apples, with each student carrying 24 to start and everyone having 6 at the end (each student giving two apples to each teacher).

And so on, though at some point one has to wonder if the students were driving to school in dump-trucks full of apples!

Shawn| PUZZLE GRANDMASTER | Profile May 9th, 2008 - 10:57 amThe answer must be (9 teachers) + (3 students) = 12 apples / student

Each student gives up 9 apples by offering each one to each teacher. Since there are 3 students, each teacher receives 3 apples.

foger1979| Profile May 9th, 2008 - 11:39 amEach girl started with 12 apples.

After they handed 1 apple each to each of the 9 teachers, that left them with 3 each. And the teachers each received 3.

michaelc| Profile May 9th, 2008 - 12:46 pm36 Apples, or any whole number multiple thereof.

You can generalize the whole thing for different scenarios.

Let S be the number of Students.

Let T be the number of Teachers.

Let A be the total number of all the folks.

Let X be the starting apples.

Let i be the iterations, (number of times 1 apple is given by each student to all the teachers)

Let n be the final number of apples help by each.

We have the following equations…

A = S + T

X = LCM (S, T, A)

where LCM is the least common multiple.

i = (X – (XS/A))/ T

n = i*S = X – (iT)

Plug the formulas into an Excel sheet, and see how many apples for any number of student/teacher combiniations!

Cool problem!

michaelc| Profile May 9th, 2008 - 3:45 pmMy bad. I need to correct my last post.

Any multipule of 12 is how many apples each student has.

The number of apples each student starts with would be just the total number of folks, and not the least common multipule. All of the other equations hold. Also any multiplule of the total number of folks is also a solution.

I get the feeling I sometimes overcomplicate things…

tad| Profile May 9th, 2008 - 4:03 pm12

Diego| Profile May 9th, 2008 - 8:18 pmThey have at first twelve times the number of apples they give

mgillig| Profile May 9th, 2008 - 10:38 pmThey each had to have 4 apples to begin with.

lil_b7717| Profile May 10th, 2008 - 3:00 pmThere iz a lot of answers that could be true…

1: 3 each

2: 6 each

3: 9 each etc.

jasc| Profile May 11th, 2008 - 12:09 ameach student had 12

3×12 = 12×3

scottk| Profile May 11th, 2008 - 10:16 am12 apples. they each have 12, give 1 to each teacher. that leaves each student with 3 apples left, and each teacher wih 3 apples

RK| Founder | Profile May 12th, 2008 - 7:34 amAs many of you figured out, the answer is 12.

Marioberges, Lograh, & Falwan provide more detailed explanations. Michaelc also has an interesting submission