# Two Value Overlay (TVO)

Although the full Pattern Overlay Method (POM) is not available in the Sysudoku solvers, a shadow of POM is offered in both. Two Value Overlay (TVO) discards patterns of one value that would eliminate the true pattern of a second value. Phillip Beeby’s Philsfolly solver offers Two Value Overlay for selection as the POMs option on its action menu. Andrew Stuarts includes it on its own solving sequence unless it is rejected.

Two Value Overlay is a 3-step process, illustrated here on the 5-panel and 3-panel from Heine’s ultrahardcore 3.

Step 1 is to find two cells such that one of them is included in every pattern of some value. This might not require identifying all patterns. Going South to North, any 7-freeform starting in r8c8 misses r8c9. Is there a 7 cell that must be included in such a freeform? r7c9 is included to give c9 a 7, and is the only 7 allowed in the West box, leaving only 7r4c3 as a 7 in r4. Thus every 7 pattern includes r4c3 or r8c9

Step 2 is to find another value Y whose candidates include the step 1 cells. Above, the 3-panel reveals that Y can be 3.

The true pattern of Y cannot include the step 1 cells, because it would then lose at least one of its candidates, a true one. So

Step 3: discard all Y patterns that include both step 1 cells and remove any resulting orphans.

But there is an alternative when identifying all Y patterns is difficult.  It may happen that Y in one step 1 cell implies Y in the other.  Then at least, that Y candidate can be removed, because its presence alone takes both step 1 cells from the true Y pattern.  7. Above, a 7 forcing chain shows that 3r4c3 => 3r8c9 so 3r4c3 can be  removed without finding all 3 patterns.

Another way to bypass the full Step 3 is shown in Weekly Extreme 427 (4/07/15).

All North to South freeforms are easily found.  Three start in r4c5 and one starting in r4c6 including r5c9.

The step 1 cells hold 6-candidates, but the full step 3 looks difficult and unproductive.  On the 6-panel, only one South to North freeform is deleted. Then it becomes apparent that many 6 candidates are confirmed when 6r5c9 is true. Perhaps the chain of confirmations will reach 6r4c5, so that 6r5c9 can be removed. A few more arrows make the case. It can be shown with forcing chains as well.

A similar bypass of the normal TVO step 3 occurs in Wex 428, posted 4/14/15.  A Beeby POMs announcement reads:

“Between them r5c3 and r7c9 include all patterns of 5, so patterns of 7 which include both cells can be deleted. As a result, no pattern of 7 includes r5c3 so 7 can be deleted from r5c3.”

Every 5 W>E missing r5c3 hits r7c9, so all 5-patterns include r5c3 or r7c9.

This time, an AIC forcing chain shows that  7r5c3 => 7r7c9,  and must be removed to allow a 5-pattern.

Yes, the 5-panel made us look, but actually no West to East 7 pattern starting in c1 can include r5c3 anyway.

The same announcement is made for Beeby’s second TVO in Wex 428:

“Between them, r2c1 and r6c2 are included in all patterns of 8, so patterns of 4 which include both cells can be deleted. As a result, no pattern of 4 includes r6c2 so 4 can be deleted from r6c2”

But steps 1 and 3 can be reasoned differently:

Step 1: The West to East patterns missing both r2c1 and r6c2 fail to reach c8, because they must cross r5c2 and r4c4, or r45c4.  One of the two cells must be included in the true 8 patterns.

Step 3:  4r6c2 => 4r2c1, and is removed

To try for more removals, you could generate freeforms missing a step 1 cell to reach any remaining 4-candidates. Escaping candidates would be step 3 orphans, were there any.

Does the full step 3 ever work? An example is Heine’s ultrahardcore 397, where Beeby announces “Between them r3c3 and r7c6 include all patterns of 3. As a result, no pattern of 6 includes r1c9.

The East to West 3-freeforms not starting in r7c6 include r3c3. We devote a 6-panel to each c9 starting cell.

The dashed ff gets across, by skipping r8.

After these removals, a finned swordfish, AIC, coloring, and DIY UR wraps remain.

The ANL / X-Panel Overlay

There is an application of the X-panel TVO practice in ANL Building.

The true pattern cannot include the terminals of an Almost Nice Loop, because one terminal is false. The true pattern does include one ANL terminal. We can discard all freeforms including both ANL terminals, and look for orphans.

In this example from KrazyDad Insane 445, Beeby finds a step 3 removal on a POMs request, because “Between them r1c5 and r6c8 include all patterns of 2”, without mentioning the ALN. The red freeforms are deleted, orphaning 3r2c7.