Repeating Patterns

Days, clock hours and the last digit of a power all run in a loop. Once you know the cycle length, reaching position 500 takes one division instead of five hundred steps.

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Days of the week, hours on a clock, the last digit of a growing power — all of them run in a loop and start over. Once you know the length of the loop, reaching position takes one division instead of five hundred steps.

Theorem The remainder is the answer
Every repeating pattern has a fixed cycle length. To find what sits at any position:

Divide the position by the cycle length, keep the remainder, then count that many places.

"After" — count the remainder forward, starting from the next item.
"Before" — count the remainder backward, starting from the previous item.
"From" or "ends on" — count the remainder starting from the item itself.

If the remainder is , the position lands on the last item of the cycle — which is the same item you started from.

Example Days of the week — cycle 7

a) Today is Wednesday. What day is it in days?

remainder
"After" — count forward from the next day: Thursday, Friday, Saturday
Saturday

b) Today is Thursday. What day was it days ago?

remainder
"Before" — count backward: Wednesday, Tuesday, Monday
Monday

c) A Hijri year begins on a Tuesday. On what day does it end?

A Hijri year has days
remainder
"Ends on" — count from the day itself: Tuesday, Wednesday, Thursday, Friday, Saturday
Saturday

A Hijri year is days and a Gregorian year . When a question does not say which calendar it means, treat it as Hijri.

Example Hours on a clock — cycle 12

It is o'clock now. What time will it be in hours?

remainder
Move forward hours:
is past , so subtract:
o'clock

The clock face repeats every hours, and therefore also every , or . Any multiple of the cycle length can be discarded without changing the answer.

Example A repeating sequence

The numbers repeat in that order for ever. What is the th number?

The cycle length is
remainder
Remainder means the second item in the cycle

Twenty-one whole cycles pass by unnoticed. Only the leftover decides the answer.

Example The units digit of a power

The units digits of the powers of cycle as well:

, , , ,
Units digits: — then it starts again, so the cycle length is

What is the units digit of ?

remainder
Remainder means the second digit in the cycle
The units digit is

Note that the power itself is astronomically large, yet its final digit costs one division to find. That is the whole point of spotting the cycle.

Summary
  1. Every repeating pattern has a fixed cycle length.
  2. Divide the position by that length and keep only the remainder.
  3. "After" counts forward from the next item; "before" counts backward; "from" counts from the item itself.
  4. A remainder of lands on the last item of the cycle.
  5. Days have cycle , clock hours cycle , and a Hijri year is days.
  6. The units digits of a power repeat too — find the cycle, then use the remainder.
283. Relations and Functions
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