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(game,)f(w)m(e)i(kno)m(w)g(that)e(it)g(cannot)h(c)m(hange)g(the)g(game) f(at)g(all,)f(b)s(ecause)324 1216 y(it)h(do)s(es)i(not)f(c)m(hange)h (the)g(set)g(of)e(mo)m(v)m(es)j(a)m(v)-5 b(ailable)31 b(to)j(either)g(pla)m(y)m(er.)49 b(Therefore)36 b(w)m(e)324 1337 y(kno)m(w)d(that)g Fe(N)10 b Ff(\()p Fe(G)p Ff(\))28 b(=)f(0.)44 b(That)33 b(is,)f(\(omitting)d Fe(N)10 b Ff(\(\)\):)1789 1537 y Fb(fjg)27 b Ff(=)h(0)324 1805 y(Actually)-8 b(,)36 b(w)m(e)i(can)e(\014nd)h(a)f(whole)g(class)h(of)f (other)g(games)g(that)h(m)m(ust)f(ha)m(v)m(e)i(v)-5 b(alue)36 b(0.)324 1925 y(Supp)s(ose)44 b(w)m(e)g(ha)m(v)m(e)h(ANY)f(game)e Fe(G)h Ff(in)g(whic)m(h)h(the)f(next)i(pla)m(y)m(er)e(to)g(mo)m(v)m(e)h (can)f(b)s(e)324 2045 y(forced)37 b(to)f(lose.)55 b(It)37 b(turns)g(out)f(that)h(w)m(e)g(m)m(ust)g(ha)m(v)m(e)h Fe(N)10 b Ff(\()p Fe(G)p Ff(\))35 b(=)f(0.)56 b(Wh)m(y?)g(Consider)324 2166 y(an)m(y)37 b(game)e Fe(H)8 b Ff(.)55 b(What)36 b(happ)s(ens)i(in)d Fe(G)25 b Ff(+)f Fe(H)8 b Ff(?)55 b(Sa)m(y)37 b(Left)f(going)f(second)j(in)d Fe(H)44 b Ff(has)37 b(a)324 2286 y(winning)d(strategy)h(in)g Fe(H)8 b Ff(.)50 b(Then)36 b(Left)f(can)h(win)e(in)h Fe(G)23 b Ff(+)h Fe(H)43 b Ff(b)m(y)36 b(seeing)f(whic)m(h)g(game)324 2407 y(Righ)m(t)30 b(mo)m(v)m(es)i(in,)e(and)h(follo)m(wing)e(the)i (winning)f(strategy)i(in)e(that)h(game.)42 b(Righ)m(t)30 b(will)324 2527 y(ev)m(en)m(tually)d(get)h(stuc)m(k)g(with)f(no)g(mo)m (v)m(es,)i(and)f(Left)f(will)e(win.)41 b(The)28 b(same)f(thing)f(w)m (orks)324 2647 y(if)k(Left)h(has)h(a)f(win)g(going)e(\014rst)j(in)f Fe(H)8 b Ff(,)31 b(or)g(Righ)m(t)f(has)i(a)e(win)h(going)f(\014rst)i (or)f(second)h(in)324 2768 y Fe(H)8 b Ff(.)50 b(Therefore,)37 b(for)e(our)g(theory)h(to)e(w)m(ork)i(w)m(e)h(kno)m(w)f(that)f(the)g (outcome)g(of)g(a)f(game)324 2888 y(is)e(not)g(c)m(hanged)i(b)m(y)f (adding)f Fe(G)h Ff(to)f(it.)42 b(This)33 b(implies)d(that)j Fe(N)10 b Ff(\()p Fe(G)p Ff(\))28 b(=)f(0.)324 3075 y(What)h(ab)s(out)g (the)h(game)f Fb(f)p Ff(0)p Fb(jg)p Ff(?)41 b(\(Note)28 b(that)h(w)m(e'v)m(e)h(used)g(\\0")e(to)g(indicate)f(a)h(game)g(of)324 3195 y(v)-5 b(alue)35 b(0,)i(the)g(simplest)e(example)h(b)s(eing)g Fb(fjg)p Ff(.\))53 b(In)37 b Fb(f)p Ff(0)p Fb(jg)p Ff(,)f(Left)g(can)h (mo)m(v)m(e)g(and)f(lea)m(v)m(e)324 3316 y(righ)m(t)k(lo)s(oking)f(at)h Fb(fjg)p Ff(,)j(and)e(Righ)m(t)f(has)h(no)g(mo)m(v)m(e)g(at)g(all.)67 b(Our)41 b(guiding)e(in)m(tuition)324 3436 y(that)j(the)g(v)-5 b(alue)42 b(should)g(b)s(e)g(the)h(n)m(um)m(b)s(er)f(of)g(mo)m(v)m(es)h (adv)-5 b(an)m(tage)42 b(to)g(left)f(indicates)324 3556 y(that)32 b(this)g(game)g(should)h(ha)m(v)m(e)g(v)-5 b(alue)32 b(1.)1765 3757 y Fb(f)p Ff(0)p Fb(jg)26 b Ff(=)i(1)324 3958 y(By)33 b(symmetry)-8 b(,)33 b(w)m(e)g(m)m(ust)g(ha)m(v)m(e)h (that:)1726 4159 y Fb(fj)p Ff(0)p Fb(g)27 b Ff(=)g Fb(\000)p Ff(1)324 4360 y(W)-8 b(e)33 b(w)m(an)m(t)g(addition)e(to)h(w)m(ork,)i (so)f(this)f(requires)h(that:)1603 4561 y Fb(f)p Ff(0)p Fb(jg)21 b Ff(+)h Fb(fj)p Ff(0)p Fb(g)27 b Ff(=)h(0)p Fe(:)324 4761 y Ff(This)36 b(is)f(of)g(course,)i(not)f(the)g(same)f (game)g(as)h Fb(fjg)p Ff(.)52 b(But)35 b(it)g(do)s(es)h(ha)m(v)m(e)h (the)f(prop)s(ert)m(y)324 4882 y(that)f(the)g(\014rst)h(pla)m(y)m(er)f (to)g(mo)m(v)m(e)g(in)f(this)h(game)f(loses.)51 b(So)35 b(ev)m(erything)h(is)f(consisten)m(t)324 5002 y(so)e(far.)1918 5251 y(2)p eop %%Page: 3 3 3 2 bop 324 548 a Ff(W)-8 b(e)27 b(can)h(con)m(tin)m(ue)g(to)f(build)f (up)i(games)f(and)g(their)g(v)-5 b(alues.)41 b(What)28 b(ab)s(out)f Fb(f)p Ff(1)p Fb(jg)p Ff(?)40 b(This)324 668 y(clearly)g(has)h(v)-5 b(alue)41 b(2,)i(b)s(ecause)g(Left)e(can)g (mak)m(e)g(t)m(w)m(o)h(mo)m(v)m(es)g(to)f(Righ)m(t's)g(zero.)69 b(In)324 789 y(general)1671 992 y Fb(f)p Fe(n)p Fb(jg)27 b Ff(=)h Fe(n)22 b Ff(+)g(1)1570 1279 y Fb(fj)g(\000)h Fe(n)p Fb(g)k Ff(=)h Fb(\000)p Fe(n)23 b Fb(\000)f Ff(1)324 1482 y(What)40 b(ab)s(out)f(the)h(game)f Fb(f)p Ff(0)p Fb(j)p Ff(1)p Fb(g)p Ff(?)64 b(What)40 b(m)m(ust)g(its)f(v)-5 b(alue)39 b(b)s(e?)65 b(\(P)m(erhaps)42 b(w)m(orking)324 1603 y(this)28 b(out)h(with)f(hac)m(k)m(en)m(bush)k(mak)m(es)d(this)g (easier)f(to)h(see,)h(but)f(w)m(e'll)f(pro)s(ceed)i(with)e(the)324 1723 y(set)33 b(notation.\))42 b(Consider)33 b(the)g(follo)m(wing)d (game:)1406 1926 y Fe(G)d Ff(=)h Fb(f)p Ff(0)p Fb(j)p Ff(1)p Fb(g)21 b Ff(+)h Fb(f)p Ff(0)p Fb(j)p Ff(1)p Fb(g)f Ff(+)h Fb(fj)p Ff(0)p Fb(g)324 2130 y Ff(What)38 b(happ)s(ens)h(if)e (Righ)m(t)h(mo)m(v)m(es)h(\014rst?)61 b(Ignoring)37 b(iden)m(tical)f (alternativ)m(es,)k(he)e(has)324 2250 y(t)m(w)m(o)33 b(options:)985 2454 y Fb(f)p Ff(0)p Fb(j)p Ff(1)p Fb(g)21 b Ff(+)h Fb(f)p Ff(0)p Fb(j)p Ff(1)p Fb(g)f Ff(+)h Fb(fjg)65 b Ff({OR{)96 b Fb(f)p Ff(0)p Fb(j)p Ff(1)p Fb(g)21 b Ff(+)h(1)g(+)g Fb(fj)p Ff(0)p Fb(g)324 2657 y Ff(That)33 b(is:)1134 2777 y Fb(f)p Ff(0)p Fb(j)p Ff(1)p Fb(g)21 b Ff(+)h Fb(f)p Ff(0)p Fb(j)p Ff(1)p Fb(g)64 b Ff({OR{)96 b Fb(f)p Ff(0)p Fb(j)p Ff(1)p Fb(g)21 b Ff(+)h(1)g(+)g Fb(\000)p Ff(1)324 2945 y(That)33 b(is:)1341 3065 y Fb(f)p Ff(0)p Fb(j)p Ff(1)p Fb(g)21 b Ff(+)h Fb(f)p Ff(0)p Fb(j)p Ff(1)p Fb(g)64 b Ff({OR{)97 b Fb(f)p Ff(0)p Fb(j)p Ff(1)p Fb(g)324 3232 y Ff(In)34 b(either)g(case,)i(Left)e(wins.)48 b(\(In)34 b(the)h(c)m(hoice)f(on)g(the)h(righ)m(t,)e(Left)h(mo)m(v)m (es)h(and)g(lea)m(v)m(es)324 3353 y(Righ)m(t)26 b(with)g Fb(fjg)p Ff(,)h(and)g(in)f(the)h(left)f(c)m(hoice)h(Left)f(mo)m(v)m(es) i(and)f(lea)m(v)m(es)h(Righ)m(t)d(with)i Fb(f)p Ff(0)p Fb(j)p Ff(1)p Fb(g)p Ff(,)324 3473 y(whic)m(h)33 b(loses)g(for)f(Righ)m (t\).)324 3660 y(No)m(w,)37 b(going)e(bac)m(k)h(to)g Fe(G)p Ff(,)h(what)f(happ)s(ens)h(when)g(Left)e(mo)m(v)m(es)i(\014rst?) 54 b(Left)36 b(has)g(only)324 3780 y(one)d(c)m(hoice)g(\(eliminating)28 b(symmetry\),)33 b(and)f(lea)m(v)m(es)i(the)f(follo)m(wing:)1558 3984 y Fb(fjg)22 b Ff(+)g Fb(f)p Ff(0)p Fb(j)p Ff(1)p Fb(g)f Ff(+)h Fb(fj)p Ff(0)p Fb(g)324 4187 y Ff(That)33 b(is:)1682 4308 y Fb(f)p Ff(0)p Fb(j)p Ff(1)p Fb(g)21 b Ff(+)h Fb(fj)p Ff(0)p Fb(g)324 4475 y Ff(No)m(w)33 b(righ)m(t)f(has)h(t)m(w)m(o)g(c)m(hoices:)1088 4678 y Fb(f)p Ff(0)p Fb(jg)22 b Ff(+)g Fb(fj)p Ff(0)p Fb(g)64 b Ff({OR{)97 b Fb(f)p Ff(0)p Fb(j)p Ff(1)p Fb(g)21 b Ff(+)h Fb(fjg)27 b Ff(=)g Fb(f)p Ff(0)p Fb(j)p Ff(1)p Fb(g)324 4882 y Ff(The)34 b(c)m(hoice)f(on)g(the)h(left)e(lets)h(Righ)m (t)f(win,)h(the)h(other)f(c)m(hoice)h(loses)f(for)f(Righ)m(t,)h(so)g (he)324 5002 y(a)m(v)m(oids)g(it.)1918 5251 y(3)p eop %%Page: 4 4 4 3 bop 324 548 a Ff(So)36 b(game)g Fe(G)g Ff(has)h(the)g(prop)s(ert)m (y)g(that)g(whic)m(hev)m(er)h(pla)m(y)m(er)f(starts)g(that)f(pla)m(y)m (er)h(loses.)324 668 y(This)30 b(means)f(it)g(m)m(ust)h(ha)m(v)m(e)h(v) -5 b(alue)29 b(0.)42 b(Let)30 b Fe(x)g Ff(b)s(e)g(the)g(v)-5 b(alue)29 b(of)h Fb(f)p Ff(0)p Fb(j)p Ff(1)p Fb(g)p Ff(.)41 b(W)-8 b(e)30 b(kno)m(w)h(that)1576 888 y Fe(x)23 b Ff(+)f Fe(x)g Ff(+)g(\()p Fb(\000)p Ff(1\))28 b(=)g(0)324 1108 y(Therefore)33 b(in)f(order)h(for)f(our)g(theory)i(to)e(w)m(ork)h(w)m (e)h(ha)m(v)m(e:)1730 1367 y Fb(f)p Ff(0)p Fb(j)p Ff(1)p Fb(g)27 b Ff(=)2096 1300 y(1)p 2096 1344 49 4 v 2096 1436 a(2)324 1613 y(W)-8 b(e)47 b(can)g(go)f(on)g(this)g(w)m(a)m(y)i (and)f(deriv)m(e)g(more)f(and)h(more)e(results)i(lik)m(e)f(this.)85 b(F)-8 b(or)324 1733 y(example,)28 b(y)m(ou)f(can)h(pro)m(v)m(e)g(that) f Fb(fj)p Ff(4)p Fb(g)g Ff(=)h(0,)g Fb(f)p Ff(3)p Fb(j)p Ff(5)p Fb(g)e Ff(=)i(4,)g Fb(f)p Ff(3)p Fb(j)p Ff(4)p Fb(g)e Ff(=)i(3)11 b(+)2929 1694 y Fc(1)p 2929 1710 36 4 v 2929 1767 a(2)2974 1733 y Ff(,)28 b(etc.)42 b(Instead,)324 1853 y(let's)30 b(just)g(state)g(the)h(rule)e(that)h(de\014nes)i(these) f(v)-5 b(alues,)30 b(and)h(pro)m(v)m(e)g(the)f(whole)g(theory)324 1974 y(w)m(orks)k(with)e(these)i(v)-5 b(alues.)324 2161 y(De\014nition)31 b(of)h(v)-5 b(alue)32 b(of)g(a)g(game:)568 2430 y(Sa)m(y)k Fe(G)e Ff(=)f Fb(f)p Fe(G)1102 2394 y Fd(L)1154 2430 y Fb(j)p Fe(G)1259 2394 y Fd(R)1316 2430 y Fb(g)p Ff(.)54 b(Recursiv)m(ely)37 b(compute)f(the)g(v)-5 b(alues)36 b(of)g(eac)m(h)g(ele-)568 2551 y(men)m(t)f(of)g Fe(G)1008 2515 y Fd(L)1060 2551 y Ff(,)h(let)f(the)g(set)h(of)f(n)m(um) m(b)s(ers)i(that)e(results)g(b)s(e)h Fe(A)p Ff(.)52 b(Similarly)-8 b(,)568 2671 y(compute)33 b Fe(B)39 b Ff(from)32 b Fe(G)1385 2635 y Fd(R)1442 2671 y Ff(.)46 b(No)m(w)34 b Fe(A)g Ff(and)f Fe(B)39 b Ff(are)33 b(sets)h(of)f(n)m(um)m(b)s(ers.)47 b(De\014ne)568 2792 y Fe(a)38 b Ff(=)g(max\()p Fe(A)p Ff(\))h(and)g Fe(b)g Ff(=)f Fe(min)p Ff(\()p Fe(B)5 b Ff(\),)41 b(with)e(the)g(con)m(v)m(en)m(tion)h(that)f(if)e Fe(A)i Ff(is)568 2912 y(empt)m(y)27 b Fe(a)h Ff(=)g Fb(\0001)p Ff(,)g(and)f(if)f Fe(B)33 b Ff(is)26 b(empt)m(y)i Fe(b)g Ff(=)g Fb(1)p Ff(.)41 b(W)-8 b(e)27 b(require)h(that)f Fe(a)h(<)f(b)p Ff(.)568 3032 y(\(Otherwise)33 b(this)f(pro)s(cedure)i (fails)d(to)h(assign)h(a)f(v)-5 b(alue)32 b(to)g Fe(G)p Ff(.\))44 b(No)m(w)33 b(the)568 3153 y(v)-5 b(alue)32 b(of)g Fe(G)g Ff(is)g(the)h(\\simplest")e(n)m(um)m(b)s(er)i(in)f(the)h (op)s(en)g(in)m(terv)-5 b(al)31 b(\()p Fe(a;)17 b(b)p Ff(\).)568 3315 y(It)30 b(remains)g(to)g(explain)g(what)h(\\simple")d (means.)43 b(The)32 b(in)m(tegers)f(are)f(the)568 3435 y(simplest)39 b(n)m(um)m(b)s(ers,)k(with)d(0)g(b)s(eing)g(the)g (simplest)f(of)h(all,)g(follo)m(w)m(ed)g(b)m(y)568 3555 y Fb(\006)p Ff(1)27 b(follw)m(ed)e(b)m(y)j Fb(\006)p Ff(2)f(etc.)42 b(After)27 b(that)g(comes)g(the)g(dy)m(adic)g(n)m(um)m (b)s(ers,)i(that)568 3676 y(is,)g(rational)e(n)m(um)m(b)s(ers)k(with)e (p)s(o)m(w)m(ers)h(of)f(t)m(w)m(o)h(in)f(the)h(denominator.)41 b(The)568 3796 y(lo)m(w)m(er)32 b(the)h(p)s(o)m(w)m(er)h(of)e(2)g(the)h (simpler)e(the)i(n)m(um)m(b)s(er.)568 3958 y([There)27 b(is)e(a)g(unique)h(simplest)f(n)m(um)m(b)s(er)h(in)f(an)m(y)h(in)m (terv)-5 b(al)25 b(\()p Fe(a;)17 b(b)p Ff(\).)41 b(Because)568 4078 y(if)35 b(an)g(in)m(terv)-5 b(al)35 b(con)m(tains)h(t)m(w)m(o)h(n) m(um)m(b)s(ers)g(of)f(the)g(same)g(simplicit)m(y)-8 b(,)34 b(then)568 4199 y(it)d(also)h(con)m(tains)g(an)h(ev)m(en)h(simpler)d(n) m(um)m(b)s(er.])324 4469 y(Note)i(that)g(this)g(de\014nition)f (requires)i(that)f Fe(a)c(<)f(b)p Ff(,)34 b(and)g(it)e(requires)i(this) f(recursiv)m(ely)324 4589 y(for)24 b(all)e(subgames)j(of)f Fe(G)p Ff(.)41 b(So,)26 b(for)e(example,)h(this)g(de\014nition)e(fails) g(to)h(assign)g(a)g(n)m(um)m(b)s(er)324 4709 y(go)32 b Fb(f)p Ff(0)p Fb(j)p Ff(0)p Fb(g)p Ff(,)g(whic)m(h)i(is)f(the)g(game) f(where)j(the)e(\014rst)h(pla)m(y)m(er)f(to)g(mo)m(v)m(e)h(wins.)45 b(This)33 b(v)-5 b(alue)324 4830 y(of)32 b(this)h(game)f(is)h(not)g(a)f (n)m(um)m(b)s(er,)i(and)f(w)m(e)h(denote)g(it)e(b)m(y)i Fb(\003)p Ff(.)45 b(W)-8 b(e'll)31 b(talk)i(more)f(ab)s(out)324 4950 y(games)g(that)g(don't)h(ha)m(v)m(e)h(n)m(um)m(b)s(ers)f(later.) 1918 5251 y(4)p eop %%Page: 5 5 5 4 bop 324 548 a Ff(No)m(w)31 b(our)g(job)f(is)h(to)f(pro)m(v)m(e)i (that)f(this)f(w)m(a)m(y)i(of)f(assigning)e(a)i(v)-5 b(alue)30 b(to)g(a)h(game)f(has)h(the)324 668 y(sum)h(prop)s(ert)m(y)-8 b(.)324 855 y(Theorem:)49 b(Let)36 b Fe(G)f Ff(and)g Fe(H)43 b Ff(b)s(e)36 b(games)f(that)g(ha)m(v)m(e)h(v)-5 b(alues,)36 b(and)g(let)f(these)h(v)-5 b(alues)35 b(b)s(e)324 976 y Fe(g)h Ff(and)c Fe(h)h Ff(resp)s(ectiv)m(ely)-8 b(.)44 b(Then)34 b(the)f(game)f Fe(G)22 b Ff(+)g Fe(H)40 b Ff(has)33 b(v)-5 b(alue)32 b Fe(g)25 b Ff(+)d Fe(h)p Ff(.)324 1162 y(Pro)s(of:)46 b(Let's)35 b(\014rst)f(lo)s(ok)f(closely)h (at)g(the)h(game)e(that)h(is)g(the)g(sum)g(of)g Fe(G)g Ff(and)h Fe(H)8 b Ff(.)47 b(Let)324 1283 y Fe(G)27 b Ff(=)h Fb(f)p Fe(G)659 1247 y Fd(L)711 1283 y Fb(j)p Fe(G)816 1247 y Fd(R)873 1283 y Fb(g)e Ff(and)g Fe(H)35 b Ff(=)28 b Fb(f)p Fe(H)1491 1247 y Fd(L)1542 1283 y Fb(j)p Fe(H)1659 1247 y Fd(R)1716 1283 y Fb(g)p Ff(.)41 b(What)26 b(are)g(the)h(left)e(sets)i(of)e Fe(G)9 b Ff(+)g Fe(H)f Ff(?)40 b(Left)26 b(can)324 1403 y(c)m(ho)s(ose)35 b(an)f(elemen)m(t)h(of)f Fe(G)1325 1367 y Fd(L)1411 1403 y Ff(and)h(lea)m(v)m(e)g Fe(H)41 b Ff(in)m(tact,)35 b(or)f(Left)g(can)h (c)m(ho)s(ose)g(an)g(elemen)m(t)324 1523 y(of)d Fe(H)524 1487 y Fd(L)608 1523 y Ff(and)h(lea)m(v)m(e)g Fe(G)f Ff(in)m(tact.)43 b(The)34 b(c)m(hoices)f(for)f(Righ)m(t)g(are)h (similar.)40 b(Th)m(us:)951 1729 y Fe(G)22 b Ff(+)g Fe(H)36 b Ff(=)27 b Fb(f)p Fe(G)1495 1688 y Fd(L)1569 1729 y Ff(+)22 b Fe(H)r(;)17 b(G)22 b Ff(+)g Fe(H)2080 1688 y Fd(L)2132 1729 y Fb(j)p Fe(G)2237 1688 y Fd(R)2316 1729 y Ff(+)g Fe(H)r(;)17 b(G)22 b Ff(+)g Fe(H)2827 1688 y Fd(R)2884 1729 y Fb(g)324 1934 y Ff(\(Where)j(b)m(y)g Fe(G)869 1898 y Fd(L)927 1934 y Ff(+)5 b Fe(H)32 b Ff(w)m(e)25 b(mean)f(the)h(union)e(of)h(\()p Fe(X)13 b Ff(+)5 b Fe(H)j Ff(\))24 b(o)m(v)m(er)i(all)c Fe(X)32 b Ff(that)24 b(are)g(elemen)m(ts) 324 2055 y(of)32 b Fe(G)512 2019 y Fd(L)564 2055 y Ff(.\))324 2242 y(Let)j Fe(g)552 2205 y Fd(L)638 2242 y Ff(and)g Fe(g)881 2205 y Fd(R)973 2242 y Ff(and)g Fe(h)1221 2205 y Fd(L)1308 2242 y Ff(and)g Fe(h)1556 2205 y Fd(R)1649 2242 y Ff(b)s(e)g(the)h(v)-5 b(alues)35 b(of)f(the)h(left)g(and)g(righ) m(t)f(sides)h(of)g Fe(G)324 2362 y Ff(and)30 b Fe(H)38 b Ff(resp)s(ectiv)m(ely)-8 b(.)44 b(That)31 b(is,)f Fe(g)1607 2326 y Fd(L)1686 2362 y Ff(=)e(max)o(\()p Fe(G)2086 2326 y Fd(L)2138 2362 y Ff(\))j(and)f Fe(g)2445 2326 y Fd(R)2530 2362 y Ff(=)d(min)o(\()p Fe(G)2911 2326 y Fd(R)2969 2362 y Ff(\),)j(etc.)44 b(Then)31 b Fe(g)324 2482 y Ff(is)h(the)g(simplest)g (n)m(um)m(b)s(er)h(in)e(\()p Fe(g)1527 2446 y Fd(L)1579 2482 y Fe(;)17 b(g)1674 2446 y Fd(R)1730 2482 y Ff(\))32 b(and)h Fe(h)f Ff(is)g(the)h(simplest)e(n)m(um)m(b)s(er)i(in)f(\()p Fe(h)3287 2446 y Fd(L)3339 2482 y Fe(;)17 b(h)3439 2446 y Fd(R)3497 2482 y Ff(\).)324 2603 y(Here's)34 b(a)e(sc)m(hematic)h (diagram)d(of)i(the)h(situation:)426 2980 y Fa (---\(------|----\)--------)q(-\(-)q(-|--)q(-\)--)q(----)q(----)q(\(--) q(-\(--)q(|---)q(\)\)--)q(-)939 3101 y(g)872 b(h)1025 b(g+h)324 3358 y Ff(The)27 b(t)m(w)m(o)g(parens)g(to)f(the)g(left)f(of) h Fe(g)13 b Ff(+)c Fe(h)25 b Ff(corresp)s(ond)i(to)f(the)h(options)e (for)h(Left:)40 b Fe(G)3327 3322 y Fd(L)3388 3358 y Ff(+)9 b Fe(H)324 3479 y Ff(\(whose)30 b(v)-5 b(alue)29 b(is)g Fe(g)1041 3442 y Fd(L)1107 3479 y Ff(+)15 b Fe(h)p Ff(\))30 b(and)f Fe(G)15 b Ff(+)g Fe(H)1780 3442 y Fd(L)1861 3479 y Ff(\(whose)31 b(v)-5 b(alue)28 b(is)h Fe(g)18 b Ff(+)d Fe(h)2739 3442 y Fd(L)2792 3479 y Ff(\).)42 b(The)30 b(ones)g(to)f(the)324 3599 y(righ)m(t)35 b(of)h Fe(g)28 b Ff(+)d Fe(h)36 b Ff(are)h(the)g(c)m(hoices)g(for)f(Righ)m(t.)54 b(\(Note)36 b(that)h(this)f(picture)g(is)g(logically)324 3719 y(correct,)d(ev)m(en)h(if)e(some)g(of)g(the)h(parens)h(are)e(lo)s (cated)g(at)g Fb(\0001)h Ff(or)f Fb(1)p Ff(.\))324 3906 y(Clearly)d Fe(g)22 b Ff(+)17 b Fe(h)31 b Ff(is)f(in)g(the)h(range)f (de\014ned)i(b)m(y)g(these)f(paren)m(theses.)46 b(W)-8 b(e)31 b(need)g(to)f(pro)m(v)m(e)324 4027 y(that)i Fe(g)26 b Ff(+)c Fe(h)32 b Ff(is)g(the)h(simplest)f(n)m(um)m(b)s(er)h(in)f (that)g(range.)324 4213 y(W)-8 b(e)33 b(break)g(the)g(pro)s(of)f(do)m (wn)h(in)m(to)f(a)g(n)m(um)m(b)s(er)h(of)f(cases)i(\(can)f(this)f(b)s (e)h(a)m(v)m(oided?\).)324 4400 y(Case)43 b(1:)62 b Fe(g)45 b Ff(and)d Fe(h)h Ff(are)f(in)m(tegers.)72 b(If)42 b Fe(g)32 b Ff(+)c Fe(h)44 b Ff(=)g(0,)g(w)m(e're)f(done,)i(b)s(ecause)f (0)e(is)f(the)324 4521 y(simplest)g(n)m(um)m(b)s(er)h(of)f(all.)70 b(If)42 b Fe(g)31 b Ff(+)e Fe(h)43 b(>)h Ff(0)e(then)g(w)m(e)h(kno)m(w) g(that)f(one)g(of)f Fe(g)k Ff(or)d Fe(h)g Ff(is)324 4641 y(p)s(ositiv)m(e,)h(so)f(assume)g(wlog)e(that)i Fe(h)h(>)g Ff(0.)70 b(Therefore)43 b Fe(h)2526 4605 y Fd(L)2619 4641 y Ff(is)f(at)f(most)g(1)g(b)s(elo)m(w)g Fe(h)p Ff(.)324 4761 y(Therefore)j Fe(h)832 4725 y Fd(L)913 4761 y Ff(+)30 b Fe(g)46 b Ff(is)c(at)h(most)g(one)g(b)s(elo)m(w)g Fe(g)32 b Ff(+)d Fe(h)p Ff(.)75 b(And)44 b Fe(g)32 b Ff(+)e Fe(h)43 b Ff(is)f(the)i(simplest)324 4882 y(n)m(um)m(b)s(er)29 b(in)f(the)h(range.)43 b(An)29 b(analogous)e(pro)s(of)h(w)m(orks)j(if)c Fe(g)18 b Ff(+)d Fe(h)27 b(<)h Ff(0.)42 b(This)29 b(completes)324 5002 y(case)k(1.)1918 5251 y(5)p eop %%Page: 6 6 6 5 bop 324 548 a Ff(Case)31 b(2:)42 b Fe(g)34 b Ff(is)29 b(an)i(in)m(teger,)f Fe(h)h Ff(is)e(a)h(dy)m(adic)h(n)m(um)m(b)s(er.)43 b Fe(g)21 b Ff(+)c Fe(h)30 b Ff(is)g(also)g(a)f(dy)m(adic)i(n)m(um)m(b) s(er,)324 668 y(with)40 b(the)h(same)f(simplicit)m(y)e(as)j Fe(h)p Ff(.)67 b(In)41 b(fact,)h(the)f(in)m(terv)-5 b(al)39 b(\()p Fe(h)2702 632 y Fd(L)2782 668 y Ff(+)27 b Fe(g)t(;)17 b(h)3036 632 y Fd(R)3121 668 y Ff(+)27 b Fe(g)t Ff(\))40 b(is)g(an)324 789 y(iden)m(tical)i(cop)m(y)k(of)e(\()p Fe(h)1185 753 y Fd(L)1237 789 y Fe(;)17 b(h)1337 753 y Fd(R)1395 789 y Ff(\))44 b(when)h(view)m(ed)h(in)d(terms)i(of)f(the)g (simplicities)d(of)j(the)324 909 y(n)m(um)m(b)s(ers)34 b(in)e(it.)45 b(Therefore)34 b(if)e Fe(h)h Ff(is)g(the)g(simplest)f(n)m (um)m(b)s(er)i(in)e(\()p Fe(h)2803 873 y Fd(L)2856 909 y Fe(;)17 b(h)2956 873 y Fd(R)3013 909 y Ff(\),)33 b(then)h Fe(g)26 b Ff(+)c Fe(h)324 1029 y Ff(m)m(ust)32 b(b)s(e)h(the)g (simplest)e(n)m(um)m(b)s(er)i(in)f(\()p Fe(g)25 b Ff(+)d Fe(h)1979 993 y Fd(L)2031 1029 y Fe(;)17 b(g)25 b Ff(+)d Fe(h)2301 993 y Fd(R)2359 1029 y Ff(\).)43 b(\(And)33 b(the)g(in)m(terv)-5 b(al)31 b(that)i(w)m(e)324 1150 y(need)25 b(to)e(consider)i(ma)m(y)e(ev)m(en)j(b)s(e)e(smaller)e({)h (it)g(m)m(ust)h(b)s(e)h(con)m(tained)f(in)f(\()p Fe(g)3039 1114 y Fd(L)3095 1150 y Ff(+)t Fe(h;)17 b(g)3326 1114 y Fd(R)3387 1150 y Ff(+)t Fe(h)p Ff(\))324 1270 y(to)s(o,)32 b(but)g(this)h(only)f(further)h(reduces)h(the)f(c)m(hoices)g(a)m(v)-5 b(ailable.\))324 1457 y(Case)36 b(3:)49 b Fe(g)38 b Ff(and)d Fe(h)h Ff(are)f(b)s(oth)g(dy)m(adic)h(n)m(um)m(b)s(ers)g(with)f Fe(g)j Ff(simpler)c(than)h Fe(h)p Ff(.)52 b(Actually)-8 b(,)324 1577 y(the)31 b(argumen)m(t)f(in)f(case)i(2)g(w)m(orks)g(here)g (to)s(o.)42 b(Because)32 b(the)f(in)m(terv)-5 b(al)29 b(\()p Fe(h)2991 1541 y Fd(L)3061 1577 y Ff(+)17 b Fe(g)t(;)g(h)3305 1541 y Fd(R)3380 1577 y Ff(+)g Fe(g)t Ff(\))324 1698 y(is)32 b(an)g(iden)m(tical)f(cop)m(y)j(of)e(\()p Fe(h)1383 1662 y Fd(L)1435 1698 y Fe(;)17 b(h)1535 1662 y Fd(R)1593 1698 y Ff(\),)32 b(as)h(describ)s(ed)g(ab)s(o)m(v)m(e.)324 1885 y(Case)c(4:)42 b Fe(g)31 b Ff(and)e Fe(h)g Ff(are)f(b)s(oth)g(dy)m (adic)h(n)m(um)m(b)s(ers)h(of)e(the)h(same)f(simplicit)m(y)-8 b(.)40 b(In)28 b(this)h(case)324 2005 y(it)i(turns)h(out)g(that)f Fe(g)24 b Ff(+)d Fe(h)32 b Ff(simpler)e(than)i Fe(g)j Ff(and)d Fe(h)g Ff(\(in)f(fact)h(it)e(could)i(b)s(e)g(an)g(in)m (teger\).)324 2125 y(No)m(w)c(the)f(in)m(terv)-5 b(al)26 b(\()p Fe(h)1144 2089 y Fd(L)1208 2125 y Ff(+)11 b Fe(g)t(;)17 b(h)1446 2089 y Fd(R)1514 2125 y Ff(+)11 b Fe(g)t Ff(\))27 b(is)g(an)g(iden)m(tical)f(cop)m(y)i(of)f(\()p Fe(h)2750 2089 y Fd(L)2802 2125 y Fe(;)17 b(h)2902 2089 y Fd(R)2960 2125 y Ff(\))27 b(when)h(view)m(ed)324 2246 y(in)39 b(terms)h(of)g(the) g(simplicities)d(of)i(the)i(n)m(um)m(b)s(ers)g(in)e(it)g({EX)m(CEPT{)j (that)e Fe(g)31 b Ff(+)c Fe(h)40 b Ff(is)324 2366 y(simpler)c(than)i Fe(g)t Ff(.)60 b(Therefore,)40 b(if)d Fe(g)k Ff(w)m(as)e(the)g (simplest)e(in)g(its)h(in)m(terv)-5 b(al,)38 b(then)h Fe(g)29 b Ff(+)c Fe(h)324 2487 y Ff(it's)32 b(certainly)g(the)h (simplest)e(n)m(um)m(b)s(er)i(in)f(an)g(ev)m(en)j(smaller)30 b(in)m(terv)-5 b(al)31 b(around)i(it.)324 2673 y(QED.)1918 5251 y(6)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF