Uniformly at Random

Archive for June 2009

The necklace splitting problem

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The necklace splitting problem may be formulated (somewhat fancifully) as follows.

Suppose that two thieves have stolen a necklace.  The necklace is open at the clasp and consists of some number of jewels of k different types.  There are an even number of jewels of each type.  The thieves wish to cut the necklace into as few pieces as possible and share the pieces between the two of them so that each thief has a fair share of each type of jewel.  What is the minimum number of cuts that suffices?

Alon and West (Proc. Amer. Math. Soc. 98 (1986) 623-628) showed that k cuts always suffices for a fair division of the necklace, regardless of the total number of jewels.  Remarkably, the (very short!) proof of this combinatorial result uses a result from topology, namely the Borsuk-Ulam theorem.

Written by uncudh

June 9, 2009 at 3:04 pm

Posted in math

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The grief of Gudrun

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Apparently, Tennyson’s poem “Home They Brought Her Warrior Dead” was inspired by a translation of the First Lay of Gudrun from the Poetic Edda. Here is the Tennyson poem:

HOME they brought her warrior dead:
She nor swooned, nor uttered cry:
All her maidens, watching, said,
‘She must weep or she will die.’

Then they praised him, soft and low
Called him worthy to be loved,
Truest friend and noblest foe;
Yet she neither spoke nor moved.

Stole a maiden from her place,
Lightly to the warrior stepped
Took the face-cloth from the face;
Yet she neither moved nor wept.

Rose a nurse of ninety years,
Set his child upon her knee—
Like summer tempest came her tears
‘Sweet my child, I live for thee.’

The Eddaic lay begins (Hollander trans.):

Erst Gjuki’s daughter     unto death was nigh
as o’er Sigurth she sate     sorrowfully;
she whimpered not,     nor her hands she wrung,
nor wept, either,     as do women else.

Went to the widow     wise earls kindly,
the heavy heart     of her to ease;
nor yet Guthrun     her grief could weep,
in her bosom through     her heart would burst.

Curiously, in the Norse lay, it is not the placing of her child on her knee that causes Gudrun to finally express her grief, but rather it is the uncovering of Sigurd’s corpse that causes her to burst into tears.

Written by uncudh

June 8, 2009 at 11:30 pm

Posted in literature

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