Clock times and calendar dates: patterns the format lets you have

0:00Somebody glances at a clock, sees four forty-four, and thinks: nice. Or ten ten. Or twelve twenty-one. Your book collects those three, and asks you to find every time of each kind. Nothing happens to time at twelve twenty-one. The second before and the second after are like any other second. The pattern is not in the time. It is in the display — in how many boxes the clock has, and what each box is allowed to hold.

0:33Which makes it countable — and that is the whole video. So look at what the clock actually offers you. Four slots. An hour, then two minute slots — and not one of them is free. The hour runs from one to twelve. That is all. The first minute slot runs from zero to five. Nothing else, because sixty minutes means its digit never reaches six. The second minute slot is the only generous one. Zero to nine.

1:06Those three ranges are the entire rulebook. Which means we do not have to hunt for pretty times. We can list them, and know when we have finished. Type one. Four forty-four. The hour digit, repeated in both minute slots. So we want h, colon, h, h. Let's try them. One eleven. Two twenty-two. Three thirty-three. Four forty-four. Five fifty-five. And then six. Six sixty-six? There is no such time. Sixty-six is not a minute reading.

1:43And that is the first minute slot doing its job. It stops at five. So type one has exactly five members, and the list is complete, because the sixth is blocked by a rule we already wrote down. Type two. Ten ten. The hour block copied into the minute block. Ten ten. Eleven eleven. Twelve twelve. Those obviously work. But what about one? Does one o'clock give you one oh one?

2:14And here it gets interesting, because it depends entirely on how your clock writes the hour. If it shows a leading zero, then oh one oh one works. So does oh two oh two, all the way up to oh nine oh nine. That is twelve times on the list. If your clock does not pad, so it says one rather than oh one, then only ten ten, eleven eleven and twelve twelve qualify.

2:42Three times, not twelve. Same clock, same instants, different list. Type three. Twelve twenty-one. The whole display reading the same in both directions. With four slots and a leading zero, here is the complete list. Oh one ten. Oh two twenty. Oh three thirty. Oh four forty. Oh five fifty. And then it stops, because oh six sixty would need a minute reading of sixty. Then ten oh one. Eleven eleven. And twelve twenty-one.

3:17Eight times in the whole twelve hours. That is the lot. And eleven eleven is on this list and the last one too. Some times are pretty in more than one way. Now let's take that leading zero seriously, because two honest people will give you two different answers here. Drop the padding, and a single-digit hour gives you three slots instead of four. One, colon, zero, one. Read it backwards: one, zero, one. That is a palindrome.

3:52So one oh one counts. And so do one eleven, one twenty-one, one thirty-one, one forty-one, one fifty-one. And the same again for two, three, all the way to nine. Nine hours, six minute options each. Fifty-four palindromic times, instead of eight. So the answer to how many pretty times there are is not eight, and it is not fifty-four. It is: tell me what your clock looks like. Which is this whole module in one line. A property of the writing is not a property of the thing.

4:31Your book asks a nice version of this. Start at ten oh one. How long until the next palindromic time? Do not search for it. Use the list. After ten oh one comes eleven eleven. Ten oh one to eleven oh one is sixty minutes. Then eleven oh one to eleven eleven is another ten. Seventy minutes. Now the one after that. Eleven eleven to twelve twenty-one. Eleven eleven to twelve eleven is sixty. Twelve eleven to twelve twenty-one is ten more. Seventy minutes again.

5:10The same wait, twice. And that is not a coincidence — each step forward adds exactly one hour and ten minutes to the display. Now dates. Your book introduces two children with remarkable birthdays. Manish was born on the twentieth of December, twenty twelve. Write it out as digits. Two zero, one two, two zero one two. Look at that. Twenty twelve, and then twenty twelve again. The day and the month together spell out the year.

5:45So how would you find another one? The day and month have to form the year, and that pins down the day completely. If the year starts with a twenty, the day has to be the twentieth. If it starts with nineteen, the day has to be the nineteenth. So the last century gave exactly twelve of these: the nineteenth of each month, from nineteen oh one to nineteen twelve.

6:12And this century gives twelve more, the twentieth of each month from two thousand and one to twenty twelve. Manish's birthday is the last of them. The next one after his is the twenty-first of January, twenty one oh one. Eighty-nine years away. His sister Meghana was born on the eleventh of February, twenty eleven. As digits: one one, zero two, two zero one one. Now read that backwards. One one, zero two, two zero one one. The same.

6:49Eight digits reading the same from either end. A palindromic date. And here is something your book does not tell you. Every single palindromic date this century falls in February. Not most of them. All of them. There are twenty-nine between two thousand and twenty ninety-nine, and every one is in February. Why? Because of the slots, again. A year in this century starts with a two and a zero. Two, zero, something, something.

7:23For the date to read the same backwards, those two leading digits have to turn up at the very end, in reverse order. Which forces the month to be zero two. Zero two is February. There is no choice in it at all. The month was decided by the century, before anybody picked a date. And the scarcity comes from the same place. The day slot stops at thirty-one and the month at twelve, so most digit patterns are not legal dates at all.

7:56A palindromic date is not rare because it is special. It is rare because the slots are tight. A third pupil, Jeevan, asks why we need a new calendar each year, and whether an old one could ever be reused. Your book asks that and does not answer it. So let's do the arithmetic. An ordinary year is three hundred and sixty-five days, and fifty-two weeks is three hundred and sixty-four. So a year is fifty-two weeks and one day left over.

8:32That leftover day is why your birthday moves forward one weekday every year. A leap year is fifty-two weeks and two days, so it shifts you by two instead. For a calendar to be reusable, those shifts must add up to a whole number of weeks, and the leap-year status has to match too. Work it through and twenty thirteen's calendar, for instance, comes back in twenty nineteen. Six years later.

9:01So yes. Keep your old calendars. Some come round again. Last, a different question, from a pupil called Pratibha. Take four digits. Four, seven, three and two. The largest arrangement is seven four three two. The smallest is two three four seven. Their difference is five thousand and eighty-five. Their sum is nine thousand seven hundred and seventy-nine. And then the book asks four things. Find digits making the difference bigger. And smaller. The sum bigger. And smaller.

9:39You could guess. Or you could find the structure, which turns guessing into control. Sort your digits: biggest, second, third, smallest. The difference always comes out as nine hundred and ninety-nine lots of the biggest minus the smallest, plus ninety lots of the second minus the third. Check it. Seven take two is five, and five nine hundred and ninety-nines is four thousand nine hundred and ninety-five. Four take three is one, worth ninety more.

10:13And that adds to five thousand and eighty-five. Exactly the printed answer. So to make the difference bigger, pull the outer two digits apart. The middle pair barely matters — they are worth ninety each, against nine hundred and ninety-nine. Next time, rearranging a sum so that you can do the whole thing in your head.