Fortune-Telling with Daisy Petals

What have we learned from the Clock Arithmetic blog to help us strategize about daisy petals?  Group size is two, because there are only two statements:  My Love loves me; My Love loves me not.  Division by two yields a remainder of 0 if the number of petals is even, and a remainder of 1 if the number of petals is odd.  With both even and odd numbers, division by 2 produces two groups with the same number of items in each group. 

According to the rules of the game, a speaker must alternate the two statements.  Daisy petals can be picked in any order, but it is natural to start at the top and move clockwise.  So how to ensure that My Love loves me is uttered when the last petal is picked?  If the daisy has an odd number of petals, the last petal will be the remainder after dividing by two, which is one; the statement associated with that petal must be the same as the statement associated with the first statement.  The last petal on a daisy with an even number of petals must be an even number, the opposite parity to petal #1.

The way to beat this game is to begin with the negative statement, “My Love loves me not”, when the daisy has an even number of petals, and the positive version when the daisy has an odd number of petals.

But what if the daisy field contains many types, and you must pick one blindfolded?  A Google search tells me that most field daisies have 13, 21 or 34 petals.  Note that these are also Fibonacci Numbers.  Since two of these numbers are odd, and only one even, odds are 2:1 in favor starting with “My Love loves me”.  Optimism is, after all, the natural inclination for one seeking romance.

Suppose you are a clever shepherd or shepherdess in 17^th century France; previous fortune-telling sessions have taught you that 13-, 21- and 34-petal daisies are equally prevalent in your fields.   Can you improve on the 2:1 odds of always beginning with “My Love loves Me”?

Consider adding a third outcome, “Maybe”, and alter the rules of the game by selecting another daisy if “Maybe” turns up.  With a group size of three as the divisor, the remainders after dividing the three varieties of daisies are 1, 0 and 1, respectively.  Start petal-plucking with “My Love loves me”, follow with “My Love loves me not”, and say “Maybe” last.  If you happen to select a 13- or a 34-petal daisy on the first try, you get the desired outcome immediately.  Otherwise, with a 21-petal daisy, you get a “Maybe” answer plus a do-over.

Eventually, you will pick a 13- or 34-petal daisy; the worst case scenario is that you exhaust the field of 21-petal daisies before getting the answer you want.  Even if you don’t alter the rules of the game to allow a “Mulligan”, this ordering of three statements precludes the terrible outcome of unrequited love.

The order in which you pick daisy petals doesn’t matter, but in this game the chant caller fixes the order of naming children in the first round.  That order has to be preserved in subsequent rounds.  Clock Arithmetic determines Who’s “It”:  the playing position where a chant ends is given by the remainder after dividing the number of words by the number of children. How can the chant-caller use this information to ensure that a friend is selected when the outcome is favorable, and a rival when the outcome is unfavorable?

Children playing this game are generally arranged in a circle or in a line.  Once playing position #1 is chosen by the chant caller, the child who will be “It” depends on the direction of rotation, clockwise or counter-clockwise for a circle, or to the left or right in the case of a line.

 A chant caller can use this discretion to disguise that the game is rigged.  All that is required for a favorable outcome is for the chant caller to place a favored child in a playing position equal to the remainder after dividing the number of chant words by the number of children. Same for a disfavored child. 

Accomplish this by starting the chant on a child where Winning Position coincides with the favored or disfavored child.

The chant caller can even recover from a mistake.  Suppose after starting “Eenie, Meanie” a caller wants it to end on position #5 rather than on position #6.  Try eliding the three words “catch a tiger” into two words, “catcha tiger”.  Adding another word to end on playing position #7 is trickier, but not impossible: instead of saying “If he hollers,…” try making a two-syllable word like “hollers” into two separate words:  “hol”  “lers”. 

This is the takeaway from Take-away games with just one active player:  those games can be gamed.  I did not know it at the time, but that is what I was trying to do in the summer of 1950.

Applying Clock Arithmetic and strategic thinking to daisy petals and nonsense rhymes makes for a long introduction to a fun amusement park game.  Blog #5, “Recreating the Original Ring Game”, describes the real life experience that prompted my question 72 year ago.

To set the stage for your first ring game session, please view “The Carousel Waltz—YouTube”.  This is a 3-minute clip from the 1956 movie, Carousel.  That film, in turn, was adapted from a celebrated Broadway musical which opened in 1945.  Note that actors don’t speak their lines; instead, glances, gestures and costumes foreshadow the drama.  The propulsive music of the Carousel Waltz sweeps you onto the ride.

Pay particular attention to carousel riders in the background during the first 15 seconds.  They are reaching out their right arms in a pantomime of ring-taking.  Those movements were more prominently featured in the Broadway show dance routine, and would have been understood by audiences of the time.

Music and motion in this video transport me back to very sweet memories.  One pedantic note:  the ride named “Mullins’ Carousel” is, strictly speaking, a merry-go-round.  But everyone can agree that “Carousel” is a much snappier title for a musical and a movie.