A Carrollian Tale of Ducks, Cakes … and the Logic That Lurks Beneath
(Eight miniature chapters—each an episode in Alice’s onward tumble through the land where numbers wear costumes and truth plays peek-a-boo. All puzzles and solutions are woven in; no formal proofs, only story-flow with every logical cog still turning.)
I.
The Five-Row Feast
Alice arrives at the Mock Turtle’s table:
ten cakes, two neat rows.
“Only four nudges, child,” the Turtle croons,
“and make me five rows of four.”
So Alice pushes a cherry cake here, a sponge there—
never more than four touches—
until a sugar-star appears:
every slice now sings in two different rows.
The Turtle applauds.
“See?” he chuckles,
“Sharing beats hoarding; overlap is the secret spice.”
II.
The Garden of Triplets
Next, nine cakes bloom on a lawn.
“But they must blossom as ten rows of three,
and you may not move a crumb,”
says the Dormouse, half-asleep in a teapot.
Alice squints. Lines, triangles, spirals—
she lets her eyes find paths instead of piles.
Soon ten silvery threads link the nine cakes—
every crumb part of three different garlands.
“Multiplicity,” yawns the Dormouse,
“is cheaper than multiplication.”
III.
The Apple Mirage
A high wall, a drifting dream.
Apples everywhere—until Alice tries to count.
The moment she whispers “one…,”
all but a solitary apple fade like soap-bubbles.
The dream itself curtsies and murmurs,
“Objects are born when eyes arrive,
and born only one at a time.”
IV.
The Stick That Lied
She finds a stout stick: two pounds heavy.
The Gryphon saws eight times, declares,
“Equal bits—four ounces each!”
Alice counts: nine pieces on the grass.
“Dear Gryphon, you cut more than you meant.
Your ounces are wishful.”
3 and ⁵⁶/₁₀₀ ounces each piece weighs;
the stick grins, split but not fooled.
V.
The Forgetful Grid
The Queen hands Alice a 3 × 3 block of letters.
“Copy it perfectly,” she commands.
Alice writes… “Wrong!”
Writes again… “Wrong!”
No matter how crisp her pen,
the letters slide—micro-pirouettes of meaning.
The Knave whispers,
“Repetition is a leaky bucket;
symbolic water drips at every pour.”
VI.
The Court of Wise Eyes
Four heralds shout a census:
7 sages: blind of both eyes.
10: blind of one.
5: sharp in both.
9: half-sighted.
The King wants a smaller court.
Alice counts ratios, not heads:
the pattern 7 : 10 : 5 : 9 is indivisible.
“Spare 31 or 62 or 93,” she advises.
“Anything else fractures the covenant.”
The King bows—numbers, not nobles, keep the peace today.
VII.
Alice and the Wandering Tables
Trying her sums again:
4 × 5 = 12, 4 × 6 = 13—
yet twenty never comes!
The Cat grins overhead:
“Your digits stay still, dear—
but your number-base marches three paces each time.
Chase ‘20’ and it will always be
twenty steps away.”
Alice laughs; the figures wink and march on.
VIII.
The Penny-Post Square
Victorian stamps—halfpennies to fivers—
nine designs and one spare twin.
“Lay them in a square,” says the Postmaster,
“every line must add to 11 ½ d.”
Alice slips a second halfpenny beneath a stout 6 d stamp:
every row, column, diagonal—balanced.
“One gentle overlap,” she notes,
“and the whole sheet finds its balance.”
The Postmaster stamps approval.
Epilogue of Eight Lessons
Overlap feeds order – share the cake, gain the star.
Reuse outruns addition – more paths need no extra crumbs.
Seeing makes being – one apple lives in one gaze.
Cut ≠ count – slicing reality warps expectation.
Copies decay – symbols leak with every echo.
Ratios rule – reduce to the hidden vector, or chaos returns.
Frames drift – digits are costumes; bases are stages.
One overlap can steady a plane – the twin halfpenny stills the grid.
With those eight charms tucked in her pocket,
Alice steps onward—
ready for ducks that debate philosophy,
cakes that converse in code,
and puzzles that watch the puzzler.
(And so are we.)
