Mathematical model

Variables

We associate a variable to each letter: s, e, n, d, m, o, r, y:

• e, n, d, o, r, y are defined wih a $[\![0-9]\!]$-domain,
• s, m are defined with a $[\![1-9]\!]$-domain.

Constraints

The first constraint to satisfy is that no two letters are assigned to the same digit:

• $\forall i,j \in \{s, e, n, d, m, o, r, y\}, i\ne j$

Since we handle the constraint “a word cannot start with à 0” directly in the variables domain, the other constraints deal with the equation itself. There are two options, either a unique scalar product, with no additional variables, or cut it up wrt columns.

Globally:

$$1000\times s + 100\times e + 10\times n + 1\times d$$ $$+ 1000\times m + 100\times o + 10\times r + 1\times e$$ $$= 10000\times m + 1000\times o + 100\times n + 10\times e + 1\times y$$

Locally

$$d + e = y + 10\times r_1$$

$$r_1 + n + r = e +10\times r_2$$

$$r_2 + e + o = n +10\times r_3$$

$$r_3 + s + e = o + 10\times m$$

where $r_1,r_2,r_3$ are $[\![0-1]\!]$-domain variables and express the carries.