← Range ArcadeWORLD Nº 053 · BELLS WITH KNOWN PERIODS
♫ the belfryPULL THE FEWEST ROPES THAT MEET THE ORDER

TWENTY BELLS. ONE REGISTER. EVERY PERIOD A POWER OF TWO.

An order comes in.
Choose the ropes.

Each rope adds a shift to the register: bell j hears bells j−1, j−3… and flips.
Pull ropes until the ordered bells ring at exactly the periods asked, with as few ropes as you can.

THE ORDERORDER 1 / 6
♫

Bells that keep
perfect time.

A rope is a shift. The periods of every bell follow from the ropes by algebra, with nothing run.
Meet the order with the fewest ropes; then ring the peal and hear it repeat.

0 POINTS1–7 PULL A ROPE · R RING · ENTER CASTRING THE CHANGES

txs.spectrum_linear_exact: (1+P)^(2^k) = 1 + P(z^(2^k)) over GF(2), so every bell's period is read off a few polynomial products, never a tick of simulation.

A DIFFERENT SET OF ORDERS

Words choose which orders arrive at the belfry.

THE BELL-FOUNDER'S NOTES

A register, some shifts, and a proven period for every bell.

The belfry runs x ← x XOR (x≪s₁) XOR (x≪s₂) … on twenty bits. Bell j only ever hears bells below it, so the top of the register can never reach down, and every bell is exactly periodic from the first tick with a period that is a power of two.

Which power is settled by algebra. Over GF(2), squaring is linear, so (1+P)^(2^k) = 1 + P(z^(2^k)); a few polynomial products say which bells still move after 2^k ticks. No ticks are ever run to find out, and the peal you hear is just the register running to confirm it.

An order names two or three bells and their periods. Many rope sets meet the same order, because the spectrum is a coarse invariant (31 rope sets over five ropes make only 13 spectra). Points: 20 for meeting the order, 15 more for the fewest ropes, less for each rope over. Three stars is six orders cast at the cheapest cost.

SIX ORDERS CAST

♫

Every bell
on its own clock.

A PEAL FOR A FRIEND

Ring these changes with me.