In this post I will show how clock arithmetic helps you to analyze “Fortune-Telling with Daisy Petals” and “Eenie, Meanie, Meinie, Moe” as take-away games.

Let’s adhere to tradition by keeping 12 at the top at the top of the clock face, and start from there. 

Where will the hour hand be pointing after 20 hours have elapsed, when the hour hand moves in a “clockwise” direction?

This is really a question about measuring distance around a circle, or, more generally, a closed curve.

That distance measure is not the same as distance along the number line.

0….5….10….15….20….

You could measure distance from 0 to 20 by counting 20 steps, or, by subtracting the starting point, 0, from the ending point, 20: 20-0=20. 

You could also count out 20 hours, steps, around the clock face, but that gets boring.

STEP 2:

Of the original 20 hours, 8 hours remain.  For our purposes, remember that we can ignore the initial 12 hours, and claim that the hour hand will be pointing to 8 o’clock after 20 hours have elapsed.

If the hour hand were initially pointing to, say, 3 o’clock, after 20 hours it would be pointing to 3+8=11 o’clock.

Here’s the secret: 12 o’clock is effectively zero hour on a 12-hour clock face, and subtracting zero from any number leaves that number unchanged.

Computing groups of 12 and focusing on the remainder reminds us of division.  Recall the procedure (algorithm) you memorized for dividing:

But what if the remainder were zero, as would be the case if 24 hours had elapsed from your starting time?

Dividing 24 by 12 yields 2 groups of 12 hours each, with zero hours left over.  A remainder of zero means you end up right where you started:

Calculating remainders is the key to winning at “Fortune-Telling with Daisy Petals” and “Eenie, Meanie, Meinie, Moe”.

• First, write out on a line the seven days of the week in order:  Monday, Tuesday, Wednesday, Thursday, Friday, Saturday and Sunday. Use a line rather than a circle, because a circle might hint at clock arithmetic.

Monday Tuesday Wednesday Thursday Friday Saturday Sunday

• Next, ask someone to pick a day of the week and a number from, say, 30-70.
• After that, suppose the volunteer proposes “Tuesday” and 52.  Challenge that person to a speed calculation test:  “Starting from Tuesday, what day of the week will it be 52 days later?” You will surprise your friend by almost immediately knowing the answer is Friday.
• This is the trick: In your head divide 52 by 7, which has remainder 3.  52/7 = 7 R3.
• Each remainder equals the number of days from the day chosen by your volunteer. Three days after Tuesday is Friday so the answer is that 52 days from Tuesday is always Friday.
• If someone had proposed 49 days, the remainder after division by 7 is 0.  49/7 = 7 (Remainder = 0). Because the remainder is 0 you don’t move from Tuesday. 49 days from Tuesday is always Tuesday

Suppose there is a smart-alec in the group who asks you what day of the week it will be 3,748 days from Tuesday.  Just ask if you can use paper and pencil to help with the counting.  Write down the division: 3,748/7= 535 remainder 3.  The answer is Friday!

This also works for going back in time:  9,651 days before Tuesday.  Do the division to obtain 1,378 remainder 5. Five days before Tuesday requires us to “subtract” five days from Tuesday, which is Thursday.  You could have obtained the same result by “adding” 2 days to Tuesday; this is because 2+5=7.

You can do the same parlor trick by ordering 12 months of the year along a line.  But don’t use clock arithmetic to compute the number of days since the day you were born— Leap Years and different days per month complicate matters.

So how does Clock Arithmetic enable strategic thinking with daisy petals and nonsense words?    Here are some hints.

• First, the Fortune-Telling problem.  The game has three rules:  (1) you must play with the number of petals on your chosen daisy:  (2) you must pluck a single petal per turn; (3) you must alternate two contradictory statements each turn (Loves me, Loves me not).  Dividing petals by a group size of 2 allows only two possible remainders, zero or one.  Zero and one are consecutive even and odd numbers.
• “Eenie, Meanie, Meinie, Moe” has 20 words; these are spread in the same order each round over a number of children.  The number of children is the group size for your division; divide the 20 words of Eenie, Meanie by the number of children, and the remainder tells you which child is “It”.

The next blog, “Some Fun with Clocks”, is just that—a fun digression.  Let your minds wander a bit before submitting winning strategies to the discussion box.  Then, consult Blog #4, “Strategizing with Daisy Petals and Childhood Chants”.