## Testing Three Students

3 students are taking an examination of N subjects. Points are awarded according to the positions in each subject they took. The person who gets 1st will get the highest points, followed by 2nd and lastly 3rd. (So let 1st=X points, 2nd=Y points, 3rd=Z points.) Student B obtained X points in English. Student A obtained 22 points total. Students B and C obtained 9 points total each. How many subjects did they take in the examination, and what are the values of X, Y, Z?

Source: Reader Jimmy Chng Gim Hong, citing a Chinese puzzle book.

Solutions were received from Joseph DeVincentis, Dave Peck, Al Zimmermann, Jeremy Galvagni, Brian Willett, Ross Millikan, Jayavel Sounderpandian, Claudio Baiocchi, Morandi Maurizio, John Hewson, Denis Borris, Walter Reid, Hagen von Eitzen, Sudipta Das, Adrian Atanasiu, Alan O'Donnell, Kitsuki Ikeda.

Jospeh DeVincentis addressed the intended and extension solutions:

```There are 40 total points, so the total points per subject X+Y+Z must be a
factor of 40. If we assume all integer and no negative points, there must be
a minimum of 3 points per subject, so the possible values are 4, 5, 8, 10,
and 20 points per subject corresponding to 10, 8, 5, 4, and 2 subjects
respectively.

There cannot be only two subjects, because A's total of 22 would mean there
were at least 11 points awarded for 1st place, but B got only 9 points.

If there were four subjects (10 points each), A's total of 22 would mean
that at least 6 points were awarded for first place.  The possibilities that
allow A a score of 22 are 6,4,0 and 7,2,1. Neither of these allows a score
of eactly 9 points, however.

If there were five subjects (8 points each), A's total of 22 would mean that
at least 5 points were awarded for first place. The possibilities that allow
A a score of 22 are 5,2,1; 6,2,0; and 7,1,0. The second case 6,2,0 does not
allow a score of 9 points, but the others do, as 5+1+1+1+1 or 2+2+2+2+1 and
as 7+1+1+0+0. Both of these give possible answers to the whole puzzle:

A: 5+5+5+5+2 B: 1+1+1+1+5 C: 2+2+2+2+1
A: 7+7+7+1+0 B: 1+0+0+7+1 C: 0+1+1+0+7

With more than five subjects, there must be zero points for third place. We
can have either 3,2,0 or 4,1,0 for 5 points per subject (8 subjects) or
3,1,0 for 4 points per subject (10 subjects). In all of these cases it's
easy to find sets of results that work, for example:

A: 3+3+3+3+3+3+2+2 B: 0+0+0+2+2+2+0+3 C: 2+2+2+0+0+0+3+0
A: 4+4+4+4+4+1+1+0 B: 0+0+0+0+0+4+4+1 C: 1+1+1+1+1+0+0+4
A: 3+3+3+3+3+3+1+1+1+1 B: 0+0+0+0+0+0+0+3+3+3 C: 1+1+1+1+1+1+3+0+0+0

Given these multiple solutions, it's most likely that a minimum of one point
for Z was intended, making the first set of results the intended answer. 5
subjects, X=5, Y=2, Z=1. This also gives you a unique assignment of all the
points: B got 5 in English and 1 in everything else, A got 2 in English and
5 in everything else, and C got 1 in English and 2 in everything else.
```

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