A fortune that splits evenly by day but not by weekday
A reformed miser wants to give away his entire fortune — a whole number of pounds — in equal daily instalments across one year, with nothing left over. But if he instead skips Sundays and gives the same fixed amount only on weekdays, exactly one pound is left over at year's end. What is the smallest fortune, in pounds, for which this works?
Reveal the answer
£19,345. Spread as £53 a day across every day of an ordinary 365-day year that begins (and, since 365 = 52×7+1, also ends) on a Sunday — giving 53 Sundays and 312 weekdays — it divides exactly; spread as £62 on each of the 312 weekdays, it leaves £1 over. Henry Dudeney set this as 'The Converted Miser,' puzzle no. 116 in Amusements in Mathematics (1917).
— Henry Ernest Dudeney, Amusements in Mathematics — 1917, puzzle no. 116
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