# Entering the star club

Inspired by this question, I created this puzzle.

I was walking down a street, I see a star club with a security guarding the entrance door.

A man approaches him, and the guard flips a middle and a thumb on him.
The man replied by showing his fore finger.
The guard allowed him to pass.

Another man approaches, the guard flips his middle and fore finger on him.
The second man replied by showing his pinkie.
The guard allowed the second man to pass.

Another man approaches, the guard flips his ring and middle finger on him.
The third man replied by showing his thumb.
The guard allowed the third man to pass.

You decide to approach the guard. He looks at you, he stick his ring and pinkie.

What do you show?

You should show your:

Forefinger. Take the fingers to be the values 0 to 4 mod 5 (any direction and transposition will work, actually, but for simplicity, let the thumb be 0 and pinky be 4) and take their integer average mod 5.

• The explanation doesn't fit for the second man. Commented Jan 2, 2017 at 15:16
• It definitely does. Middle = 2. Fore = 1. 1+2 = 3. The part that might have confused you is that 3/2 is 4 mod 5 (since 4*2 = 8 = 3 mod 5), hence the pinky. Commented Jan 2, 2017 at 15:19
• I do not understand your explanation. What I know is 3/2 = 1.5 and 4 mod 5 = 4. But your answer "forefinger" is right. So I upvote you. I still wait another understandable explanation. Commented Jan 2, 2017 at 15:42
• Modulus division is a bit tricky to understand. Hopefully this link helps. cs.brown.edu/courses/cs007/modmult/node2.html I do Olympiad mathematics so I may have assumed wrongly that it would make sense to others (as it was the first thought I had as soon as I saw your question). If it helps, use thumb = 1 and count upwards, and take Mod residue before you divide, in this way you don't have to worry about multiplicative inverses as you only get even residues. Commented Jan 2, 2017 at 15:52

The star club is using star for their password

You have to show the finger exactly in the middle of the two fingers
thumb + fore -> ring
thumb + middle -> fore
thumb + ring -> pinkie
thumb + pinkie -> middle
fore + middle -> pinkie
fore + ring -> middle
fore + pinkie -> thumb
middle + ring -> thumb
middle + pinkie -> ring
ring + pinkie -> fore

• That's equivalent to TGE's answer - his just uses mathematics to describe "middle".
– Deusovi
Commented Jan 3, 2017 at 4:56