Multiple nice loop, short for multiple inference nice loop, is a derivative of the nice loop described in the bilocation/bivalue plot threads here and here.

As discussed, each nice loop represents a double implication chain in which each node implies only the immediate node downstream. These are nice loops in their simplest form and can be specifically referred to as simple nice loop.

So far, 2 derivatives of the simple nice loop have been identified, including the strong nice loop (grouped inference nice loop) described by Carcul here and the multiple inference nice loop to be discussed under this thread.

Drawing of the bivalue/bilocation plot and subsequently identification of simple nice loops is a non-T&E technique, but it suffers the shortcoming that an inference can only be made to the next node downstream. Multiple nice loops allow multiple inferences to be made for all nodes downstream by taking branches. This causes the nice loop to become a net (rather than chain), but it has the advantage that in effect all nodes downstream within the network can be instantly updated enabling more deductions to be made.

Each multiple nice loop represents a poly-implication chain. The additional implications are due to the extra inferences made by each node of the chain in additional to the next node downstream.

A multiple nice loop resembles the kinds of forcing net that a human is likely to identify through pure cell by cell inspection, except that the b/b plot takes away the ‘guessing’ element as the same set of nice loop propagation rule still applies.

From the following example, it can be seen that the nice loop highlighted is not a simple nice loop since [r1c6] is not a bivalue cell. However, if a multiple inference can be made at node [r1c7], then [r1c6] would become a bivalue node of {4,5} with the 9 eliminated. The result is a multiple nice loop which would eliminate a 2 in [r4c5].

Nice loop notation:

[r4c5]=3=[r4c9]=6=[r4c8]=8=[r5c7]=9=[r1c7](-9-[r1c6])=7=[r3c7]-7-[r3c6]-4-[(r1c6)]-5-[r5c6]-2-[r4c5] => r4c5<>2

where

[r1c7](-9-[r1c6])=7=[r3c7] denotes that r1c7 infers not only r3c7, but also r1c6.

The extra bracket within the node -4-[(r1c6)]-5- denotes that r1c6 is associated with a multiple inference from upstream.

Proof:

r1c7=9 => r3c7=7 & r1c6=45 => r3c6<>7 => r3c6=4 => r1c6=5 => r5c6=2 => r4c5<>2

r1c7<>9 => r5c7=9 => r4c8=8 => r4c9=6 => r4c5=3

Therefore r4c5<>2 => r4c5=3

Since one additional inference has been made with r1c7=9, this multiple nice loop is equivalent to a triple implication chain in which a bivalue tripod can be drawn cantered at r1c6.

Here is an example of a grouped x-cycle with a multiple inference.

[r4c9]-6-[r56c7](=6=[r6c4]-6-[r5c5])=6=[r7c7]-6-[r7c5|(r5c5)]=6=[r1c5]-6-[r2c6]=6=[r2c9]-6-[r4c9] => r4c9<>6

where:

[r56c7] is a grouped node.

[r56c7](=6=[r6c4]-6-[r5c5]) is a multiple inference.

[r7c5|(r5c5)]=6=[r1c5] is a strong inference due to the multiple inference on r5c5 from upstream.

Proof:

r56c7=6 => r4c9<>6

r56c7<>6 => r6c7<>6 => r6c4=6 => r5c5<>6

r56c7<>6 => r7c7=6 => r7c5<>6 (with r5c5<>6 from above) => r1c5=6 => r2c6<>6 => r2c9=6 => r4c9<>6

Therefore, r4c9<>6

You may express this multiple nice loop in terms of a triple implication chain. Since r1c5<>6 implies r5c5=6 or r7c5=6, the trilocation triangle is at [r1c5][r5c5][r7c5]

r1c5=6 => r2c6<>6 => r2c9=6 => r4c9<>6

r5c5=6 => r6c4<>6 => r6c7=6 => r4c9<>6

r7c5=6 => r7c7<>6 => r56c7=6 => r4c9<>6

Therefore, r4c9<>6