# How can I find which sport/hobby corresponds to which friend when a set of given clues is provided?

Marina, Mike, Roger, Phillip and Justin has each only one hobby from the following list: motocross, badminton, swimming, athletics and karate, but not necessarily in that order. It is known that:

• Who does swimming and Marina don't know each other.
• Justin needs a vehicle for his hobby.
• The one who does karate and Roger are friends since childhood.
• Mike is a relative of the one who does athletics who in turn is friends with Phillip.
• The one who does badminton is a friend of Phillip and of the one who does martial arts.

Who is the person that does athletics and what is Mike's hobby?.

The alternatives given in my book are as follows:

$$\begin{array}{ll} 1.&\textrm{Roger and badminton}\\ 2.&\textrm{Mike and atletics}\\ 3.&\textrm{Phillip and badminton}\\ 4.&\textrm{Marina and karate}\\ \end{array}$$

Can someone help me with this problem using the approach of logic grids? I attempted building one and I have this:

$$\begin{array}{|c|c|c|c|c|c|}\hline \textrm{person}&\textrm{motocross}&\textrm{badminton}&\textrm{swimming}&\textrm{athletics}&\textrm{karate}\\\hline \textrm{Marina}&x&&x&&\\\hline \textrm{Mike}&x&&x&x&\\\hline \textrm{Roger}&x&&x&&x\\\hline \textrm{Phillip}&x&x&\checkmark&x&x\\\hline \textrm{Justin}&\checkmark&x&x&x&x\\\hline \end{array}$$

Which leads me to decide between these two apparently valid choices:

$$\begin{array}{|c|c|c|c|c|c|}\hline \textrm{person}&\textrm{motocross}&\textrm{badminton}&\textrm{swimming}&\textrm{athletics}&\textrm{karate}\\\hline \textrm{Marina}&x&x&x&x&\checkmark\\\hline \textrm{Mike}&x&\checkmark&x&x&x\\\hline \textrm{Roger}&x&x&x&\checkmark&x\\\hline \textrm{Phillip}&x&x&\checkmark&x&x\\\hline \textrm{Justin}&\checkmark&x&x&x&x\\\hline \end{array}$$

and

$$\begin{array}{|c|c|c|c|c|c|}\hline \textrm{person}&\textrm{motocross}&\textrm{badminton}&\textrm{swimming}&\textrm{athletics}&\textrm{karate}\\\hline \textrm{Marina}&x&x&x&\checkmark&x\\\hline \textrm{Mike}&x&x&x&x&\checkmark\\\hline \textrm{Roger}&x&\checkmark&x&x&x\\\hline \textrm{Phillip}&x&x&\checkmark&x&x\\\hline \textrm{Justin}&\checkmark&x&x&x&x\\\hline \end{array}$$

The source of confusion is that there are three people for whom I can't decide which is their hobby, as the three choices left are valid for each group. The first answer appears in the alternatives as "Roger and badminton". The second answer appears also in the alternatives as "Marina and karate".

How can I decide between them both? Can someone help me how to link the other clues given?

• Those clues look incomplete. "Mike is a relative from the one who does atletics, who happens to" - who happens to what? "The one who does badminton is Phillip's friend and from the one who does martial arts." - what does "from" mean here? Are you missing some words? – Rand al'Thor Mar 15 at 20:02
• @Randal'Thor Yes there were missing words. I fixed it. Now should it be better understood. – Chris Steinbeck Bell Mar 15 at 20:34

• From the fourth clue, the one who does athletics is friends with Phillip.

• From your deductions so far, you know Phillip is the one who does swimming. That means, from the first clue, Phillip and Marina don't know each other.

• Therefore Marina can't be the one doing athletics, because that one is Phillip's friend and she isn't. This excludes your second possibility.

That should be enough to give you the unique solution. Just for completeness, though, let's go through the full deduction step by step from the beginning.

1. Who does swimming and Marina don't know each other.

At first this just tells us Marina does not do swimming, but we'll come back to this clue later.

2. Justin needs a vehicle for his hobby.

This is a roundabout way of saying Justin does motocross.

3. The one who does karate and Roger are friends since childhood.

At first this just tells us Roger does not do karate, but we'll come back to this clue later.

4. Mike is a relative of the one who does athletics who in turn is friends with Phillip.

At first this just tells us neither Mike nor Phillip does athletics, but we'll come back to this clue later.

5. The one who does badminton is a friend of Phillip and of the one who does martial arts.

This tells us Phillip does not do badminton or karate.

Now we have:

$$\begin{array}{|c|c|c|c|c|c|}\hline \textrm{person}&\textrm{motocross}&\textrm{badminton}&\textrm{swimming}&\textrm{athletics}&\textrm{karate}\\\hline \textrm{Marina}&\times&&\times&&\\\hline \textrm{Mike}&\times&&&\times&\\\hline \textrm{Roger}&\times&&&&\times\\\hline \textrm{Phillip}&\times&\times&&\times&\times\\\hline \textrm{Justin}&\checkmark&\times&\times&\times&\times\\\hline \end{array}$$

Now the only thing left for Phillip to do is swimming. After we know Phillip does swimming, the first clue becomes

Phillip and Marina don't know each other.

From the fourth and fifth clues, the ones doing athletics and badminton are both friends with Phillip. So Marina can't be one of those two, which means

Marina does karate.

$$\begin{array}{|c|c|c|c|c|c|}\hline \textrm{person}&\textrm{motocross}&\textrm{badminton}&\textrm{swimming}&\textrm{athletics}&\textrm{karate}\\\hline \textrm{Marina}&\times&\times&\times&\times&\checkmark\\\hline \textrm{Mike}&\times&&\times&\times&\times\\\hline \textrm{Roger}&\times&&\times&&\times\\\hline \textrm{Phillip}&\times&\times&\checkmark&\times&\times\\\hline \textrm{Justin}&\checkmark&\times&\times&\times&\times\\\hline \end{array}$$

And now it's easy to fill in that

Roger does athletics and Mike does badminton,