A classic Christmas story, slightly bent
In the little flat above the grocer’s, Christmas was approaching with the calm assurance of a creditor. Though poor in the usual sense, each one of its young occupants had a personal treasure: Della her hair, which would have made the Queen of Sheba jealous, and Jim his gold watch, which had been his father’s and his grandfather’s before him. Each, for all its current beauty, awaited its ideal mate: the watch, a chain, and the hair, in due course, a set of combs.
No pawnbroker could have appraised these treasures at their true worth. In the small ledger of their household, each was worth \(v\) as it stood, and 1 full unit of happiness when joined to its ideal companion. The only question was what worth Jim and Della would place on the other’s happiness, and whether it would be enough.
Jim and Della each knew the intensity of their love, and their private gain from completing the other’s treasure. These sums \(v_i\) for \(i\in\{J,D\}\) had been independently drawn from the standard uniform distribution, \(v_i\sim U[0,1]\), but neither knew the figure etched upon the other’s heart. Each must choose, without consultation or the gift of prophecy, whether to keep the treasure in hand or sacrifice it to complete the other.
\[\begin{array}{c|cc}& \text{Jim keeps} & \text{Jim sacrifices} \\ \hline \text{Della keeps} & v,v & 1,v_J \\\text{Della sacrifices} & v_D,1 & 0,0\end{array}\]
Our beautiful heroes are on equal footing, and the cream is skimmed just as certainly as Newton’s laws: each will sacrifice for the other when the value \(v_i\) exceeds \(V\), determined by fate and a balancing of nature’s forces.
Oh Della, sweet Della, who places value \(v_D\) on Jim’s happiness. If she keeps her flowing locks, she gains \(v\) if Jim, too, keeps his treasure to himself. If not, her flowing curls have found their ideal mate, realizing their full worth 1, \[vV+(1-V).\] And if she sacrifices them, the beautiful auburn strands blowing in the wind, her sacrifice delivers her \(v_D\) if Jim does not undertake the same; otherwise, her selfless act is entirely in vain, \[v_DV.\] Oh Della, sweet Della, give in to the better angels of your nature when your love is strong enough, \[v_DV\geq vV+(1-V)\Leftrightarrow v_D\geq v+\frac{1-V}{V}.\] And, our Jim, the kindest of souls, his perspective is no different. Nature harmonizes when \[V = v+\frac{1-V}{V},\] and by the power of Bhaskara’s formula, a balance arises, \[V=\frac{\sqrt{(1-v)^2+4}-(1-v)}{2}.\]
Nature smiled, pleased with her design. When each treasure has no value absent the beloved’s sacrifice, \(v=0\), her forces balance at the reciprocal of the golden ratio, the divine number, governing the growth of shells, the spread of clouds, the petals on a sunflower; when instead, each treasure needs no mate to achieve its highest value, \(v=1\), no scope for sacrifice exists. In between, the balance between forces increases \(V\), as surely as the sun rises in the East. Between these extremes, our beautiful heroes sacrifice their treasures whenever the power of their love is strong enough.
And they loved. They loved as Orpheus loved Eurydice, as Héloïse loved Abelard, as the river loves the sea. Della, sweet Della, sold her hair to buy a chain for Jim’s prized watch, and Jim, kind delightful Jim, had sold the watch to buy the combs that would smooth her flowing locks.
The magi, you may know, were wise men-wonderfully wise men-who brought gifts to the Babe in the manger, thereby beginning the tradition and the art of giving Christmas presents. And herein I have lamely related to you the uneventful chronicle of two poor fools in a flat above a grocer’s who sacrificed for each other the greatest treasures of their house. “Unlucky,” a passerby might say, “what a pity,” “what a shame.” But in a last word, let it be said that of all who give gifts these two are the luckiest. Of all of us, these foolish children are the richest. Everywhere they are the wisest. They are the magi.