# Sylver Coinage: The Progression Hypothesis

## The Hypothesis

Following Roberts, let ```a[0], a[1], ..., a[s]``` be an increasing arithmetic progression with successive difference `d`. Assume that `a[0]` and `d` are coprime and greater than 1. Considering `S = {a[0],...,a[s]}` as a position in Sylver Coinage, we can also assume that `s<a[0]`, since for any `k`, ```a[k+a[0]] = a[k] + da[0]```, so that `a[0]` and `a[k]` together eliminate `a[k+a[0]]`.

Let `t = t(S)`. The value of `t` is given by Roberts's Formula. This is the Progression Hypothesis:

• If `s=1` then `S` is an ender as shown by the proof of Hutchings's Theorem. In fact any position satisfying the conditions of Hutchings's Theorem is a quiet ender.
• If `a[0]=3` and `s=2` then `S` is a non-quiet ender.
• If `a[0]>3` and `s=a[0]-1`, then `t=sd`, and the moves that fail to eliminate `t` are the values of `kd` for all `k` less than or equal to `s` that do not divide `s`. Since in particular `s-1` does not divide `s`, `S` is not an ender.
• If `a[0]>3` and `s<a[0]-1`, then the moves that fail to eliminate `t` are the values of `t-kd` for `k` from 1 to `(a[0]-2) mod s` inclusive. If `s` divides `(a[0]-2)`, then `S` is a quiet ender; otherwise it is not an ender.

## Examples

Let `S={2,9}`. Since `a[0]=2`, `S` is a quiet ender.

Let `S={3,7,11}`. Since `a[0]=3` and `s=2`, `S` is a non-quiet ender.

Let `S={5,6,7,8,9}`. Since `a[0]>3` and `s=a[0]-1`, `S` is not an ender. Here `t=4`, `s=4`, and `d=1`; and 4 fails to be eliminated by the one multiple of 1 that does not divide `s`: 3.

Let `S={7,11,15,19,23,27,31}`. Again `a[0]>3` and `s=a[0]-1`, so `S` is not an ender. Here `t=24`, `s=6`, and `d=4`; since neither 4 nor 5 divides 6, 24 fails to be eliminated by 16 and 20.

Let `S={6,13,20,27}`. In this case `a[0]>3` and `s<a[0]-1`. We find that `t=41`, `s=3`, and `d=7`. Since `s` leaves a remainder of 1 when divided into `a[0]-2`, 41 fails to be eliminated by `(41-1*7)=34`.

Let `S={8,11,14,17,20,23,26}`. Again `a[0]>3` and `s<a[0]-1`. Here `t=29`, `s=6`, and `d=3`. Since `s | a[0]-2`, `S` is a quiet ender.

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Col. G. L. Sicherman [ HOME | MAIL ]