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zang: (( ))=    

fu: ( )( )=( )

Puzzle Example 12ed2

<#yypuz 5><q,k,m,s 2><n: *><y: {m,s}{k,m}{m}{k,m,s}>auth 12ed2</yy>

Put the outcome catalog into columns. The pool of names—k,m,q,s—has 24 = 16 combinations, and only those yielding Yang on evaluating the hidden yy-gram are given explicity. The rest yield yiN, as the catalog says—<n:*>—so we reconstruct them for listing in our yiN column.

—some unseen yy-gram—

yy col heads
—whole-pool catalog of outcomes—

{k}
{s}
{ }
{k,s}
{k,q}
{q}
{k,m,q,s}
{k,m,q}
{q,s}
{k,q,s}
{m,q,s}
{m,q}

{m,s}
{k,m}
{m}
{k,m,s}

Group the outcomes into pairs where you detect a deep or possibly bogus name. For example, the pair { }, {s} shows that s being present or absent does not effect the outcome when other names are unchanged in the two valuations.

A similar pair is {k,m,q,s}, {k,m,q}.

Among the yiN outcomes are more pairs where we see no effect—that is, different outcome—due to s, namely

{q}, {q,s} and {s}, {m,s}

Let's make a fresh catalog listing with pairs differing only in s:

—some unseen yy-gram—

yy col heads
—re-catalog to expose s

{k,s}, {k}
{s}, { }
{q,s}, {q}
{k,m,q,s}, {k,m,q}
{k,q,s}, {k,q}
{m,q,s}, {m,q}
---

{m,s}, {m}
{k,m}, {k,m,s}
---

Pairing throughout the catalog to expose s reveals no case where s is material to the outcome. Thus a solution yy-gram for this catalog need not involve s.

Since we don't need it, we can dismiss s and re-catalog:

—some unseen yy-gram—

yy col heads
—re-catalog to dismiss s

{k}
{ }
{q}
{k,m,q}
{k,q}
{m,q}
---

{m}
{k,m}
---

A glancing inspection of the yang ( ) outcomes shows that k is immaterial when m is the only other name present. It will be of interest to know about k's effect in other valuations.

—some unseen yy-gram—

yy col heads
—re-catalog to expose k

{k}, { }
{k,q}, {q}
{k,m,q}, {m,q}
---

{m}, {k,m}
---

The re-cataloguing has revealed k to be bogus. So clarify the catalog to dismiss that name.

—some unseen yy-gram—

yy col heads
—re-catalog to dismiss k

{ }
{q}
{m,q}
---

{m}
---

We might interpret the clarified catalog of outcomes in logic terms:

"(For a yang outcome the necessary and sufficient condition is:)
m AND NOT q"

Seen this way, transcribe into AND form, superenclosing q for the negation:

((m)((q)))

The yy-gram has an equivalent with fewer enclosures when we apply the flip-flop theorem "((x)) = x"

((m)q)

This solution might appear in tweet format as

<#yygram 5> <ans: ((m)q) > auth 12ed2</yy>

or as

<#yygram 5> <ans: (q(m)) > auth 12ed2</yy>

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©2012 David Zethmayr
update 2012.4.10