If no number was chosen by more than 3 people and more people chose the number 5 than the number 4, then in the worst scenario (least amount of people chose the number 1) the number 4 was chosen 2 times and the number 2, 3 and 5 were chosen 3 times. That makes a total of 3*3+2 = 11, so 1 person choose the number 1. This was the worst scenario, that guarantees that at least 1 person choose the number 1.
