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\centerline{\bf The Hints Promised in the Postscript to
  \tenbxsl Surreal Numbers}

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\textindent{9.}
To prove that $x\le x$ for all numbers $x$ in the new system, first prove
the auxiliary lemma
$$\hbox{if}\qquad x<Y_R \qquad\hbox{and}\qquad
                  y \not\le x \qquad\hbox{then}\qquad x<y,$$
by induction on $d(x)$, for all fixed $y$.

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\textindent{19.}
Perhaps the simplest answer is
$p=\bigl(\bigl\{(\{1\},\{1\})\bigr\},\{-1\}\bigr)$.
Conway has proved in fact that we can solve the equation $p+p=x$ for any
given $x$ by taking $p=\bigl(\{q\},\{x-q\}\bigr)$ where
$$q=\bigl(X_{LL}\cup(x+x-X_{RR})\cup(x+X_L-X_R),\emptyset\bigr).$$

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\textindent{22.}
Considering just the members of $S$ created before $\aleph$ day, the
unlike numbers are precisely $g_0$, $g_1$, $g_2$, $g_3$, \dots, where
$$g_n=\bigl(\{g_0,\ldots,g_{n-1}\},\,\{g_0,\ldots,g_{n-1}\}\bigr)\,.$$
Moreover, the elements $\{g_0,\ldots,g_{n-1}\}$ form a finite field (under
Conway's addition and multiplication operations), for $n=2$, 4, 16, 256,
65536, \dots, $2^{2^k}$,~\dots~!

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