What Day of the Week Will Your 100th Birthday Be?

Count in weeks

Here are some trips forward in time, all starting on a Friday. Look at the middle column.

Days later	Whole weeks and days left over	The day
7	7 × 1 + 0	Friday
10	7 × 1 + 3	Monday
14	7 × 2 + 0	Friday
30	7 × 4 + 2	Sunday
50	7 × 7 + 1	Saturday
100	7 × 14 + 2	Sunday
365	7 × 52 + 1	Saturday
1,000	7 × 142 + 6	Thursday

The whole weeks don’t matter at all: each one ends on a Friday again. Only the days left over move you on. This is Grade 6’s division: when you divide 100 by 7, the quotient is 14 (the whole weeks) and the remainder is 2 (the days left over), and 100 = 7 × 14 + 2. The remainder is always one of 0, 1, 2, 3, 4, 5 or 6, so there are only seven answers, one for each day.

A calendar page is a table of remainders

Look at a month on a calendar. Each column is one day of the week, and going down a column you add 7 each time.

So a calendar page has been doing remainders all along. Two dates fall on the same day of the week exactly when they are a whole number of weeks apart, that is, when the days between them divide by 7.

The clock on the wall

You already do this kind of arithmetic with a clock. It is 9 o’clock, and you have a 5-hour bus ride. You don’t arrive at 14 o’clock on a 12-hour clock: 9 + 5 = 14, and after 12 the hand starts again, so you arrive at 2 o’clock. The hand only ever shows what is left over after whole turns of 12 hours.

A week is a clock with seven numbers instead of twelve. Counting like this, where you go round and round and keep only what is left over, is called modular arithmetic, or clock arithmetic. The number you go round by (12 for the hours, 7 for the days, 24 for a 24-hour clock) is the modulus.

Mathematicians write

$$1{,}000 \equiv 6 \pmod 7$$

and read it as “1,000 is congruent to 6, modulo 7”. It means that 1,000 and 6 leave the same remainder when divided by 7, or, saying the same thing another way, that 1,000 − 6 = 994 is a whole number of sevens (994 = 7 × 142). On the week’s clock, 1,000 days and 6 days take you to the same place.

Why you can add and multiply remainders

Think of each number of days as whole weeks plus some days left over. Put two trips together, and the whole weeks add up to more whole weeks, which change nothing; only the days left over add up to something that matters. So to add, you can add the remainders and take the remainder of that.

For example, 1,000 days and then 30 days more: the remainders are 6 and 2, and 6 + 2 = 8 = 7 + 1, so 1,030 days is 1 day on. (Check: 1,030 = 7 × 147 + 1.)

Multiplying works for the same reason, since multiplying is adding again and again. Three trips of 1,000 days: 3 × 6 = 18 = 7 × 2 + 4, so 3,000 days from a Friday is 4 days on, a Tuesday. You never had to divide 3,000 by 7.

Six forward is one back

There is one more shortcut. On the week’s clock, going 6 days forward lands on the same day as going 1 day back, since 6 + 1 = 7 is a whole week. So 6 ≡ −1 (mod 7). That is the quickest way to see the opening answer: 1,000 days from a Friday is 6 days on, the same as 1 day back, a Thursday. In the same way 5 forward is 2 back, and 4 forward is 3 back.

One day on, every year

An ordinary year has 365 days, and 365 = 7 × 52 + 1: 52 whole weeks and 1 day more. So your birthday moves one day on in the week every year. If your birthday was on a Friday in 2025 (31 October 2025 was), it is on a Saturday in 2026 and a Sunday in 2027.

But a leap year has 366 days, with a 29 February added, and 366 = 7 × 52 + 2. Each time your birthday’s year crosses a 29 February, it moves one day more. 31 October 2028 comes after 29 February 2028, so in 2028 that birthday jumps from Sunday straight to Tuesday, skipping Monday.

Which years are leap years? The rule of the calendar we use, the Gregorian calendar, is:

1. A year that divides by 4 is a leap year (2024, 2028, 2032…),
2. except a year that divides by 100 (1900, 2100, 2200…), which is not,
3. except a year that divides by 400 (2000, 2400…), which is a leap year after all.

So 2000 was a leap year, but 2100 won’t be. The rule keeps the calendar in step with the seasons: it gives 97 leap years in every 400, an average year of 365.2425 days, very close to the time the Earth takes to go round the Sun from one spring to the next.

So here is the rule for any number of years: your birthday moves on by the number of years, plus one more for every 29 February crossed. Then take the remainder by 7.

Your 100th birthday

Put in your own birthday, and slide to 100 years.

For anyone born after 29 February 2000, as everyone at school today was, the 100 years to their 100th birthday cross 24 of the 29 Februaries (there are 25 years divisible by 4 in the way, but 2100 is not a leap year). So the birthday moves

$$100 + 24 = 124 = 7 \times 17 + 5$$

days on: 17 whole weeks and 5 days, and 5 forward is 2 back. Your 100th birthday falls two days earlier in the week than the day you were born. Born on a Thursday, as someone born on 31 October 2013 was? Your 100th birthday, 31 October 2113, is a Tuesday.

Your grandparents, if they were born in the 1950s, are luckier: their hundred years cross 2000, which is a leap year, so they cross 25 of them. 100 + 25 = 125 = 7 × 17 + 6, which is 1 back: their 100th birthday falls one day earlier in the week than the day they were born.

Your family’s 2026 wall calendar starts on a Thursday and has no 29 February. In which year could you hang it up again, with every date on the right day?

You need a year that also starts on a Thursday and is also not a leap year. Each year the start moves one day on, two after a leap year. From 2026 to 2037 is 11 years, and they cross three 29 Februaries (2028, 2032 and 2036): 11 + 3 = 14, exactly two weeks. And 2037 is not a leap year. So the 2026 calendar comes back in 2037, then in 2043, 2054, 2065, 2071, 2082, 2093 and 2099.

Between 1901 and 2099 every calendar comes back after 28 years at the latest: 28 years cross exactly 7 leap days, and 28 + 7 = 35 = 7 × 5, five whole weeks. But 2100 breaks the pattern, since it is not a leap year.

The whole calendar repeats every 400 years

Here is a surprise. The Gregorian calendar’s 400 years have 400 × 365 + 97 = 146,097 days, and

$$146{,}097 = 7 \times 20{,}871.$$

Four hundred years is a whole number of weeks, with nothing left over. So the calendar of 2026 will be exactly the calendar of 2426, every date on the same day; the dates repeat themselves, weekdays and all, every 400 years.

That has an odd result. Since the pattern is the same every 400 years, you can count how often the 13th of a month falls on each day of the week in one cycle, once and for all. The answer is not the same for every day:

Day	Monday	Tuesday	Wednesday	Thursday	Friday	Saturday	Sunday
13ths in 400 years	685	685	687	684	688	684	687

The 13th is a little more likely to be a Friday than any other day. It could never have come out even: 400 years have 4,800 months, and 4,800 is not a multiple of 7.

In 1973 the mathematician John Conway made a method, the Doomsday rule, for finding the day of the week of any date in your head. It rests on one surprising fact: in any year, these dates all fall on the same day of the week, which Conway called that year’s doomsday.

• 4 April, 6 June, 8 August, 10 October and 12 December (the even months: day = month),
• 9 May and 5 September, 11 July and 7 November (the pairs swap: the 9th of the 5th month, the 5th of the 9th),
• the last day of February (28 February, or 29 in a leap year), and so 14 March, 14 days later,
• 3 January (4 January in a leap year).

Check it with your calendar: in 2026, all of them are Saturdays. The reason is clock arithmetic again: from 4 April to 6 June is 63 days, 9 whole weeks; from 6 June to 8 August is 63 days too; and so on.

Then you need the year’s doomsday. For the years 2000 to 2099:

1. Take the year’s last two digits, and divide by 12: for 2026, 26 = 12 × 2 + 2.
2. Count how many fours fit in that remainder: 2 holds 0 fours.
3. Add the three numbers: 2 + 2 + 0 = 4.
4. Count that many days on from Tuesday (the doomsday of 2000): Tuesday + 4 = Saturday.

Why does this work? Each year moves the doomsday one day on, and each leap year one more, as with your birthday. Twelve years move it 12 + 3 = 15 days, which leaves 1: so each 12 years count as 1 (step 1’s quotient), the years left over count 1 each (its remainder), and the leap years among them 1 more each (step 2).

Now any date of 2026. Christmas, 25 December: the nearest doomsday is 12 December, a Saturday. 25 is 13 days later, and 13 = 7 + 6, so Saturday + 6, or Saturday − 1: Friday. Nepal’s New Year’s Day, Baisakh 1, 2083, was 14 April 2026: 4 April was a Saturday, and 14 is 10 days later, 10 = 7 + 3, so Tuesday.

The method has rested on one thing: the days of the week march on, one after another, without a break. Have they always?

Dates were skipped, days were not

The Gregorian calendar replaced an older one, the Julian calendar, whose rule (every fourth year a leap year, no exceptions) had let the dates drift away from the seasons. To put them back, Pope Gregory XIII’s calendar of 1582 simply left out ten dates: Thursday 4 October 1582 was followed by Friday 15 October 1582. Britain and its empire changed only in 1752, by then 11 dates behind: Wednesday 2 September 1752 was followed by Thursday 14 September. The dates jumped, but the days of the week went on in order, Thursday then Friday, Wednesday then Thursday. So counting in weeks still works across the change; counting in dates does not. (A story says that people in Britain rioted, shouting “Give us our eleven days!”; historians have found no evidence of it, and think it is very likely a myth.)

A country that skipped a Friday

In 2011 the Pacific island country of Samoa moved itself to the other side of the International Date Line, to be on the same day as Australia and New Zealand, its neighbors in trade. Thursday 29 December 2011 was followed by Saturday 31 December. That year Samoa had no Friday 30 December at all. Counting days across that night in Samoa, the rule of this topic gives the wrong day: in Samoa, the week itself was broken.

Birthdays in Bikram Sambat

Many families in Nepal keep birthdays by the Bikram Sambat calendar. Its months follow the Sun: a month is the time the Sun takes to pass through one of the twelve signs of the zodiac, and the Sun doesn’t take the same time for each. From 2000 to 2083 BS, Nepal’s months had 29, 30, 31 or 32 days (Asar has had 31 or 32, Poush 29 or 30), and the same month can have different lengths in different years.

Counting in weeks still works: a Bikram Sambat year has 365 or 366 days, so a BS birthday also moves one or two days on each year. But which years have 366 days follows the Sun, not a rule you can do in your head. From 2000 to 2083 BS they came four years apart, except twice, when they came after three (2023 to 2026, 2054 to 2057). And the BS years’ extra days don’t fall where the Gregorian 29 Februaries do, so a BS birthday wanders between dates: Kartik 14 was 31 October in 2070 BS, but 30 October in 2073, 2077 and 2081 BS. So someone born on Kartik 14, 2070 (a Thursday, 31 October 2013) can’t work out the weekday of a BS 100th birthday from a rule: it needs the calendar of 2170 BS, worked out from the Sun’s path.

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