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If I could get help with this problem, it would be greatly appreciated. I have been trying using Venn diagram, but can't seem to understand it with four circles.

On a soccer team there are four positions: goalie, back, midfielder, and forward. Each player on the team often plays more than one position during the course of the season. On one soccer team there are 16 members. All of the goalies also play forward, and one goalie plays back as well. No goalies plays midfielders . There are as many backs as midfielders. The total numbers of midfielders is two-thirds the total number of forwards. Half of the team plays midfielder. No one plays only back. Three people play three positions. Three people play only one position. The number of midfielders who play forward but not back is equal to the number of midfielders who play back but not forward and is equal to the number of midfielders who don't play anything else.

How many people play goalies? How many people play back and forward but nothing else?

I numbered the clues out to make it easier..

  1. On a soccer team there are four positions: goalie, back, midfielder, and forward.
  2. Each player on the team often plays more than one position during the course of the season.
  3. On one soccer team there are 16 members.
  4. All of the goalies also play forward, and one goalie plays back as well.
  5. No goalies plays midfielders .
  6. There are as many backs as midfielders.
  7. The total numbers of midfielders is two-thirds the total number of forwards.
  8. Half of the team plays midfielder.
  9. No one plays only back.
  10. Three people play three positions.
  11. Three people play only one position.
  12. The number of midfielders who play forward but not back is equal to the number of midfielders who play back but not forward and is equal to the number of midfielders who don't play anything else.
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  • $\begingroup$ I recommend you draw several copies of a Venn diagram for four sets, carefully label, and see how it goes. I found three nice versions for four sets at en.wikipedia.org/wiki/… , first is three circles and a banana, next is four ellipses, finally two rectangles, a circle and a peanut, by someone named Edwards. Oh, also, you cannot make a correct Venn diagram with four perfect circles; they show that also. $\endgroup$ – Will Jagy Apr 23 '14 at 17:44
  • $\begingroup$ Tried a couple of times, I do not believe your conditions are consistent, even if we allow fractional players. The problem is the large number of regions (out of 15 inside the ellipses) that must be assigned the number $0.$ $\endgroup$ – Will Jagy Apr 23 '14 at 19:07
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I checked this Venn diagram, it works, all combinations accounted for:

enter image description here

for each ellipse, the ratio of major axis to minor axis is about 1.6, all equal, one pair side by side parallel axes, then the other pair rotated exactly $90^\circ.$

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