A school bows 900 times a day: every pupil to every other, and all to the teacher
At a small school there are exactly twice as many girls as boys. Every morning, every pupil bows to every other pupil (a bow each way between any two children) and to the teacher, who bows to no one. The total comes to exactly 900 bows each morning. How many boys are at the school?
Reveal the answer
Ten boys and twenty girls — thirty pupils in all. Any two pupils exchange a bow in each direction, giving 30×29 = 870 pupil-to-pupil bows, plus 30 more bows from each pupil to the teacher, for 900 total. Henry Dudeney set it as puzzle No. 98 in Amusements in Mathematics (1917).
— Henry Ernest Dudeney, Amusements in Mathematics — 1917, Puzzle No. 98, 'Academic Courtesies'
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