← Range ArcadeWORLD Nº 017 · A RACE THE PRIMES RUN
⚑ mod 8 derbyBET ON THE NEXT TWENTY-FOUR PRIMES

EIGHT LAPS. TWENTY-FOUR PRIMES A LAP. ONE OLD BIAS.

Two sides.
One of them usually wins.

Primes are 1, 3, 5, or 7 mod 8. Fold them with XOR 5 and a running total
tells you who is ahead. Bet on the next lap.

THE DERBYLAP 1 / 8
⚑

The 5,7 side
is usually ahead.

Every prime pulls the rope one step: 1,3 mod 8 one way, 5,7 the other.
Each lap folds in twenty-four more. Bet on which side gains, or on a dead heat.
Ten chips a lap. Dead heats pay five to one.

100 CHIPS1 2 3 TO BETOFF THEY GO

race = (baller − primeSum + n − 6) / 4: the residues are never taken, the fold already knows.

ANOTHER STRETCH OF PRIMES

Words pick where in the primes the derby starts.

FORM GUIDE

Why one side keeps winning.

Fold a prime with XOR 5: it becomes p+3 if p is 1 or 3 mod 8, and p−5 if p is 5 or 7. Sum the folds and the pile comes out as the plain sum of primes, minus n−1, plus four times the race, plus five. So the race between the two halves can be read off the total.

Each lap folds in twenty-four more primes and the rope moves by the difference. Bet 1,3 gains, 5,7 gains, or dead heat; the sides pay even money, a dead heat pays five to one.

The 5,7 side leads about ninety-five per cent of the time: 5 and 7 are non-residues mod 8 and 1 is a residue, and Chebyshev's bias sends the non-residues ahead. Three stars is finishing forty chips up.

THE DERBY IS RUN

⚑

You bet
with the bias.

A LITTLE DERBY FOR A FRIEND

Come to the derby with me.