Four Ways Back to One
Meditations on Oneness · Local folio mark 11111
Rook's split dial
At the Observatory, Rook tests a dial whose positions are numbered 0 through 14. To square a position, multiply it by itself and keep only the remainder after division by 15.
We write when and have the same remainder after division by .
The return condition
Complete the folio: find every dial position for which
Since , every solution must also satisfy
Make the two local records
Complete and mark both tables on this page.
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Complete both square-remainder tables. On each dial, circle every input whose square lands at 1.
on the 3-dial 0 1 2 Remainder of on the 5-dial 0 1 2 3 4 Remainder of
Page 2 · Recombine the clues
Four possible pairs
Rook's run record gives each position one remainder on the 3-dial and one on the 5-dial. Every return must choose one circled remainder from each smaller dial.
Complete the return map
Use the table here. Use separate paper for the proof.
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Complete the four rows. In each row, find every from 0 through 14 having both stated remainders. Then list all solutions of .
Remainder of mod 3 Remainder of mod 5 from 0 through 14 1 1 1 4 2 1 2 4 -
Check every solution by writing for a whole number . Then explain why the four-row table proves that there are no other solutions.