#math trivia for #September1: #244 is the integer after (8/12)*365. Which other months start on the “right” day-number for the month?

— Burt Kaliski Jr. (@modulomathy) September 2, 2013

The “right” day-number for month number M to start on, based on the first sentence, is apparently the integer after ((M-1)/12)*365. The logic is that each month should have the same number of days, i.e., 365/12 days, so that (M-1) months have (M-1)*(365/12) = ((M-1)/12)*365 days. But we know that day-numbers are integers, so the day-number of the first day of the Mth month should be the integer after ((M-1)/12)*365. (This is for a non-leap year.)

Given this definition, the months that start on the “right” day-number are January, August, September, October, November, and December:

Month | Length | Actual Days in Previous Months | Actual Starting Day-Number | Estimated Days in Previous Months | “Right” Starting Day-Number |

January | 31 | 0 | 1 | 0 | 1 |

February | 28 | 31 | 32 | 30.42 | 31 |

March | 31 | 59 | 60 | 60.83 | 61 |

April | 30 | 90 | 91 | 91.25 | 92 |

May | 31 | 120 | 121 | 121.67 | 122 |

June | 30 | 151 | 152 | 152.08 | 153 |

July | 31 | 181 | 182 | 182.5 | 183 |

August | 31 | 212 | 213 | 212.92 | 213 |

September | 30 | 243 | 244 | 243.33 | 244 |

October | 31 | 273 | 274 | 273.75 | 274 |

November | 30 | 304 | 305 | 304.17 | 305 |

December | 31 | 334 | 335 | 334.58 | 335 |

The start of February is a day late by these measurements because January is 31 days long, when it should be 30.42 days. But the start of March is a day early because February is only 28 days long. The lag continues through July, then starting with August, the numbers catch up.

This is all based on the assumption that the “right” day-number is the integer after the estimated number of days that the prior months “should” have, where that number itself doesn’t need to be an integer. If instead we were to round the estimate to the nearest integer and add one, the situation would be a little different. In particular, we’d find that the lag continues through August, and that October and December also have a lag. The only months with “right” starting days would be January, September, and November.

Leap years could do better if the number of days per month were arranged differently. With 366 days in the year, each month “should” have 30.5 days. Using the rounding method, January should have 31, February 30, March 31, and alternating so on through the year. But because of the unusual arrangement with February having only 29, and August having 31, the only “right” months in a standard leap year (using the rounding method) are again January, September, and November. Using the next-integer method, the months that work in a leap year are January, June, and August through December.