## 5.385. sum_cubes_ctr

Origin

Arithmetic constraint.

Constraint

$\mathrm{𝚜𝚞𝚖}_\mathrm{𝚌𝚞𝚋𝚎𝚜}_\mathrm{𝚌𝚝𝚛}\left(\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂},\mathrm{𝙲𝚃𝚁},\mathrm{𝚅𝙰𝚁}\right)$

Synonyms

$\mathrm{𝚜𝚞𝚖}_\mathrm{𝚌𝚞𝚋𝚎𝚜}$, $\mathrm{𝚜𝚞𝚖}_\mathrm{𝚘𝚏}_\mathrm{𝚌𝚞𝚋𝚎𝚜}$, $\mathrm{𝚜𝚞𝚖}_\mathrm{𝚘𝚏}_\mathrm{𝚌𝚞𝚋𝚎𝚜}_\mathrm{𝚌𝚝𝚛}$.

Arguments
 $\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}$ $\mathrm{𝚌𝚘𝚕𝚕𝚎𝚌𝚝𝚒𝚘𝚗}\left(\mathrm{𝚟𝚊𝚛}-\mathrm{𝚍𝚟𝚊𝚛}\right)$ $\mathrm{𝙲𝚃𝚁}$ $\mathrm{𝚊𝚝𝚘𝚖}$ $\mathrm{𝚅𝙰𝚁}$ $\mathrm{𝚍𝚟𝚊𝚛}$
Restrictions
 $\mathrm{𝚛𝚎𝚚𝚞𝚒𝚛𝚎𝚍}$$\left(\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂},\mathrm{𝚟𝚊𝚛}\right)$ $\mathrm{𝙲𝚃𝚁}\in \left[=,\ne ,<,\ge ,>,\le \right]$
Purpose

Constraint the sum of the cubes of a set of domain variables. More precisely, let $𝚂$ denote the sum of the cubes of the variables of the $\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}$ collection (when the collection is empty the corresponding sum is equal to 0). Enforce the following constraint to hold: $𝚂\mathrm{𝙲𝚃𝚁}\mathrm{𝚅𝙰𝚁}$.

Example
$\left(〈1,2,2〉,=,17\right)$

The $\mathrm{𝚜𝚞𝚖}_\mathrm{𝚌𝚞𝚋𝚎𝚜}_\mathrm{𝚌𝚝𝚛}$ constraint holds since the condition ${1}^{3}+{2}^{3}+{2}^{3}=17$ is satisfied.

Typical
 $|\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}|>1$ $\mathrm{𝚛𝚊𝚗𝚐𝚎}$$\left(\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}.\mathrm{𝚟𝚊𝚛}\right)>1$ $\mathrm{𝙲𝚃𝚁}\in \left[=,<,\ge ,>,\le \right]$
Symmetry

Items of $\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}$ are permutable.

Arg. properties
• Contractible wrt. $\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}$ when $\mathrm{𝙲𝚃𝚁}\in \left[<,\le \right]$ and $\mathrm{𝚖𝚒𝚗𝚟𝚊𝚕}\left(\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}.\mathrm{𝚟𝚊𝚛}\right)\ge 0$.

• Contractible wrt. $\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}$ when $\mathrm{𝙲𝚃𝚁}\in \left[\ge ,>\right]$ and $\mathrm{𝚖𝚊𝚡𝚟𝚊𝚕}\left(\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}.\mathrm{𝚟𝚊𝚛}\right)\le 0$.

• Extensible wrt. $\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}$ when $\mathrm{𝙲𝚃𝚁}\in \left[\ge ,>\right]$ and $\mathrm{𝚖𝚒𝚗𝚟𝚊𝚕}\left(\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}.\mathrm{𝚟𝚊𝚛}\right)\ge 0$.

• Extensible wrt. $\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}$ when $\mathrm{𝙲𝚃𝚁}\in \left[<,\le \right]$ and $\mathrm{𝚖𝚊𝚡𝚟𝚊𝚕}\left(\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}.\mathrm{𝚟𝚊𝚛}\right)\le 0$.

• Aggregate: $\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}\left(\mathrm{𝚞𝚗𝚒𝚘𝚗}\right)$, $\mathrm{𝙲𝚃𝚁}\left(\mathrm{𝚒𝚍}\right)$, $\mathrm{𝚅𝙰𝚁}\left(+\right)$.