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zang: (( ))=    

fu: ( )( )=( )

Puzzle Example 12ed3

<#yypuz 5><d,f,p,h 1-2><n: {p,d,f,h}{p,d,h}{p,f,h}{p,h}><y:*>auth 12ed3</yy>

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

—some unseen yy-gram—

yy col heads
—whole-pool catalog of outcomes—

{p,d,f,h}
{p,d,h}
{p,f,h}
{p,h}

{d,h}
{p,d}
{d}
{f,h}
{p,f}
{p}
{f}
{h}
{ }
{p,d,f}
{d,f,h}
{d,f}

Group the outcomes into pairs where you suspect a deep or possibly bogus name. Just looking at the yiN outcome column it is clear that f is immaterial—bogus—in two pairs:

{p,d,f,h},{p,d,h}
{p,f,h}, {p,h}

showing that, in these valuation cases, f being present or absent does not effect the outcome when the other names are present.

This observation prompts us to run a re-catalog of pairs where only f changes:

—some unseen yy-gram—

yy col heads
—re-org catalog to expose f

{p,d,f,h}, {p,d,h}
{p,f,h}, {p,h}
---

{p,d,f}, {p,d}
{d,f}, {d}
{f,h}, {h}
{p,f}, {p}
{f}, { }
{d,f,h}, {d,h}
---

The exposition re-cataloging shows that f never affects the outcome. Dismiss it from further consideration by re-cataloging without it.

—some unseen yy-gram—

yy col heads
—re-org catalog to dismiss f

{p,d,h}
{p,h}
---

{p,d}
{d}
{h}
{p}
{ }
{d,h}
---

Inspection now discloses some case pairs where d is immaterial—makes no difference—so we do a pair analysis exposition on d:

—some unseen yy-gram—

yy col heads
—re-org catalog to expose d

{p,d,h}, {p,h}
---

{p,d},{p}
{d}, { }
{d,h}, {h}
---

The exposition catalog shows that d is indeed bogus. Accordingly, we re-catalog without it.

—some unseen yy-gram—

yy col heads
—re-org catalog to dismiss d

{p,h}

{h}
{p}
{ }

The clarified catalog shows a symmetry useful to recognize: role symmetry. The roles of the two active names are identical. Such symmetry will show in the yy-gram.

It is convenient, from logic habit, to state a satisfaction condition for a yiN outcome. We shall do that first, and then, to form an equivalent yang condition, reverse it by either supplying or removing a comprehensive enclosure:

For a yiN outcome the necessary and sufficient condition is p AND h:

((p)(h))

Thus, for a yang ( ) outcome the necessary and sufficient condition is

(p)(h)

Another way to arrive at the same form from the clarified catalog is to say

    for a yang ( ) outcome the necessary and sufficient condition is that p OR h be absent—NOT p . The valuation { }—both absent—NOT p OR NOT h—is covered as well since we intend inclusive-OR by default, XOR being the digital engineer's "exclusive-OR"

The solution yy-gram might appear in tweet format as

<#yygram 5> <ans: (p)(h) > auth 12ed3</yy>

or as

<#yygram 5> <ans: (h)(p) > auth 12ed3</yy>

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©2012 David Zethmayr
update 2012.4.11 - 12gb