take the larger portion). So, Joan would
accumulate only 5/8 + 1/2 = 9/8 = 1 1/8.
Joan realizes she is not obligated to
divide up the two kilograms into two
equal piles. Clearly, she does not want
them to be too unequal. For example,
if A were tiny, then Marie would simply
not choose between A1 and A2 and get
1/2 of B yielding her nearly 1 kilogram.
That would leave Joan with approximately the same amount of gold dust.
Question: Can she do better by dividing them in some other way?
Solution: However, suppose Joan divides the two so that A is 2/3 kilograms
and B is 4/3 (or 1 1/3) kilograms. Then
Joan divides A into A1 consisting of all
2/3 kilograms and A2 consisting of one
piece of dust.
pile B into B1 consisting of 1/2 kilogram and B2 also consisting of 1/2
kilogram. No matter which one Marie
chooses, Marie will get 1/4 +1/2, leaving Joan with 1 1/4.
Question: Prove Joan cannot do any
better if the two piles are equal.
Solution: To see intuitively that Joan
cannot do any better, suppose she
divides up pile A into more unequal
portions, say 7/8 in A1 and 1/8 in A2.
In that case, Marie takes A1 and Joan
receives only 1 1/8 in total. Suppose
Joan divides pile A into less unequal
portions, say 5/8 in A1 and 3/8 in A2.
In that case, Marie declines to choose
among A1 and A2. Joan then takes A1
but now Joan must divide up B into
equal portions (otherwise Marie will
WHEN THEIR PARENTS die in a tragic
accident, daughters Joan and Marie
read their parents’ Last Will and Testament. The Will is very short, because
there is only one asset: two kilograms
of gold dust.
The Will states that Joan, as the elder sister, should divide the dust into
two piles: we will call those piles A and
B. Then she is to cut pile A into two
smaller piles that we will call A1 and
A2. Marie can decide to choose one of
A1 or A2 or not. If Marie chooses, then
Joan takes the other smaller pile and
all of pile B. If Marie does not choose
between A1 and A2, then Joan can
choose one of them (presumably the
larger one) and give Marie the other
one and then Joan must cut pile B into
B1 and B2 and Marie can choose which
one she wants.
Joan and Marie, though clever mathematicians, have never gotten along.
Each wants as much gold dust as possible while obeying the rules of the Will.
Warm-Up: Suppose Joan divides
the two kilograms equally (as depicted
in the figure), so pile A weighs one kilogram as does B. How much can Joan
be sure to receive as part of her inheritance no matter how clever Marie is?
Solution to Warm-Up: If piles A
and B each weighs 1 kilogram, then
Joan can guarantee to get 1 1/4 kilograms of gold dust. Here is how: she
divides the A pile into subpile A1 consisting of 3/4 kilogram and subpile
A2 consisting of 1/4 kilogram. If Marie chooses A1, then Joan gets A2 and
all of B, thus 1 1/4 kilograms. If Marie
declines to choose either A1 or A2,
then Joan gives Marie A2 and divides
Considering willful approaches to a golden opportunity.
DOI: 10.1145/3356582 Dennis Shasha
[CONTINUED ON P. 103]
How can I
divide these piles
to obtain as much
gold as possible,
if Marie can choose