## 5.156. first_value_diff_0

Origin

Paparazzi puzzle

Constraint

$\mathrm{\pi \pi \pi \pi \pi }_\mathrm{\pi \pi \pi \pi \pi }_\mathrm{\pi \pi \pi \pi }_\mathtt{0}\left(\mathrm{\pi  \pi °\pi },\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }\right)$

Synonyms

$\mathrm{\pi \pi \pi \pi \pi }_\mathrm{\pi \pi \pi \pi \pi }_\mathrm{\pi \pi \pi \pi }_\mathrm{\pi \pi \pi \pi }_\mathtt{0}$, $\mathrm{\pi \pi \pi \pi \pi }_\mathrm{\pi \pi \pi \pi \pi }_\mathrm{\pi \pi \pi \pi \pi \pi \pi \pi \pi }_\mathrm{\pi \pi \pi \pi }_\mathtt{0}$.

Arguments
 $\mathrm{\pi  \pi °\pi }$ $\mathrm{\pi \pi \pi \pi }$ $\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }$ $\mathrm{\pi \pi \pi \pi \pi \pi \pi \pi \pi \pi }\left(\mathrm{\pi \pi \pi }-\mathrm{\pi \pi \pi \pi }\right)$
Restrictions
 $|\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }|\beta ₯1$ $\mathrm{\pi \pi \pi \pi \pi \pi \pi \pi }$$\left(\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi },\mathrm{\pi \pi \pi }\right)$
Purpose

$\mathrm{\pi  \pi °\pi }$ is equal to the first non-zero variable of the collection $\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }$.

Example
 $\left(8,β©0,0,8,0,5βͺ\right)$ $\left(4,β©4,0,8,0,5βͺ\right)$
Typical
 $|\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }|>1$ $\mathrm{\pi \pi \pi \pi \pi \pi }$$\left(\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }.\mathrm{\pi \pi \pi }\right)<0\beta ¨$$\mathrm{\pi \pi \pi ‘\pi \pi \pi }$$\left(\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }.\mathrm{\pi \pi \pi }\right)>1$ $|\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }|-$$\mathrm{\pi \pi \pi \pi \pi }_\mathrm{\pi \pi \pi \pi }_\mathtt{0}$$\left(\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }.\mathrm{\pi \pi \pi }\right)\beta ₯1$ $\beta \left(\begin{array}{c}|\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }|\beta €4,\hfill \\ |\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }|-\mathrm{\pi \pi \pi \pi \pi }_\mathrm{\pi \pi \pi \pi }_\mathtt{0}\left(\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }.\mathrm{\pi \pi \pi }\right)>1\hfill \end{array}\right)$
Arg. properties

Functional dependency: $\mathrm{\pi  \pi °\pi }$ determined by $\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }$.

Counting
 Length ($n$) 2 3 4 5 6 7 8 Solutions 8 63 624 7775 117648 2097151 43046720

Number of solutions for $\mathrm{\pi \pi \pi \pi \pi }_\mathrm{\pi \pi \pi \pi \pi }_\mathrm{\pi \pi \pi \pi }_\mathtt{0}$: domains $0..n$

Length ($n$)2345678
Total8636247775117648209715143046720
 Parameter value

14211561555196082995935380840
24211561555196082995935380840
3-211561555196082995935380840
4--1561555196082995935380840
5---1555196082995935380840
6----196082995935380840
7-----2995935380840
8------5380840

Solution count for $\mathrm{\pi \pi \pi \pi \pi }_\mathrm{\pi \pi \pi \pi \pi }_\mathrm{\pi \pi \pi \pi }_\mathtt{0}$: domains $0..n$

FigureΒ 5.156.1 depicts an automaton that only accepts all the solutions to the $\mathrm{\pi \pi \pi \pi \pi }_\mathrm{\pi \pi \pi \pi \pi }_\mathrm{\pi \pi \pi \pi }_\mathtt{0}$ constraint. This automaton uses a counter in order to record the value of the first non-zero variable ${\mathrm{\pi  \pi °\pi }}_{i}$ already encountered. To each variable ${\mathrm{\pi  \pi °\pi }}_{i}$ of the collection $\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }$ corresponds a 0-1 signature variable ${S}_{i}$. The following signature constraint links ${\mathrm{\pi  \pi °\pi }}_{i}$ and ${S}_{i}$: .