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Let $\mathfrak o$ be the ring of integers of a totally real number field. If$f$ is a quadratic form over $\mathfrak o$ and $g$ is another quadratic formover $\mathfrak o$ which represents all proper subforms of $f$, does $g$represent $f$? We show that if $g$ is indefinite, then $g$ indeed represents$f$. However, when $f$ is positive definite and indecomposable, then thereexists a $g$ which represents all proper subforms of $f$ but not $f$ itself.Along the way we give a new characterization of positive definite decomposablequadratic forms over $\mathfrak o$ and a number-field generalization of thefiniteness theorem of representations of quadratic forms by quadratic formsover $\mathbb Z$ which asserts that given any infinite set $\mathscr S$ ofclasses of positive definite integral quadratic forms over $\mathfrak o$ of afixed rank, there exists a finite subset $\mathscr S_0$ of $\mathscr S$ withthe property that a positive definite quadratic form over $\mathfrak o$represents all classes in $\mathscr S$ if and only if it represents all classesin $\mathscr S_0$.

Can we recover an integral quadratic form by representing all its subforms?