If a, b, c and d are four positive real numbers such that $\mathrm{abcd}=1$, what is the minimum value of $(1+\mathrm{a})(1+\mathrm{b})(1+\mathrm{c})(1+\mathrm{d})$ ?

In some code, letters $a,b,c,d \text{and} e$ represent numbers $2,4,5,6 \text{and} 10$. We don't know which letter represents which number. Consider the following relationships:

$\left(i\right) a+c=e$

$\left(ii\right) b-d=d$

$\left(iii\right) e+a=b$

Which statement below is true?

Each of the numbers ${\mathrm{x}}_1,{\mathrm{x}}_2,\dots ,{\mathrm{x}}_{\mathrm{n}},\mathrm{n}\ge 4$, is equal to $1$ or $-1$. Suppose

${\mathrm{x}}_1{\mathrm{x}}_2{\mathrm{x}}_3{\mathrm{x}}_4+{\mathrm{x}}_1{\mathrm{x}}_2{\mathrm{x}}_3{\mathrm{x}}_4{\mathrm{x}}_5+{\mathrm{x}}_3{\mathrm{x}}_4{\mathrm{x}}_5{\mathrm{x}}_6+\dots +{\mathrm{x}}_{\mathrm{n}-3}{\mathrm{x}}_{\mathrm{n}-2}{\mathrm{x}}_{\mathrm{n}-1}{\mathrm{x}}_{\mathrm{n}}{\mathrm{x}}_1+{\mathrm{x}}_{\mathrm{n}-1}{\mathrm{x}}_{\mathrm{n}}{\mathrm{x}}_1{\mathrm{x}}_2+{\mathrm{x}}_{\mathrm{n}}{\mathrm{x}}_1{\mathrm{x}}_2{\mathrm{x}}_3=0$, then: