= Solution
A <balanced incomplete block design> has $t$ treatments and $b$ blocks, each containing $k$ distinct treatments with $1<k<t$. Every treatment occurs in $r$ blocks and every distinct treatment pair occurs together in $\lambda>0$ blocks. A <symmetric balanced incomplete block design> has $b=t$, which implies $r=k$ by the first counting identity below.
For the separate even-order condition, let $N$ be the square treatment-by-block <incidence matrix of a set system> of a symmetric design. Its diagonal overlap counts are $r$ and its off-diagonal counts are $\lambda$, giving
$$
NN^T=(r-\lambda)I+\lambda J.
$$
The counting identities proved in the next slots show that $r=k$ and $r+(t-1)\lambda=r^2$. Consequently the <eigenvalues> of $NN^T$ are $r^2$ on the constant vector and $r-\lambda$ on its $(t-1)$-dimensional orthogonal complement. Taking <determinants> gives
$$
\det(N)^2=r^2(r-\lambda)^{t-1}.
$$
If $b=t$ is even, the exponent $t-1$ is odd. The exponent of every prime in $\det(N)^2$ and in $r^2$ is even; therefore its exponent in the positive integer $r-\lambda$ must be even. This proves the <even-order symmetric design square obstruction>:
$$
\boxed{r-\lambda\text{ is a perfect square}.}
$$
It is a necessary condition, not a sufficiency claim.
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