A Gardner's Workout: Training the Mind and Entertaining the by Martin Gardner

By Martin Gardner

Listed below are forty-one items, which have been formerly released in a number of educational journals and well known magazines, via the esteemed grasp of mathematical video games and puzzles who wrote the clinical American video games column for 25 years. those articles span quite a lot of subject matters, together with dialogue of why a working laptop or computer will regularly beat a human participant at video games of likelihood, tiling puzzles, desktop and calculator "magic" tips, and a arguable severe assessment of a instructing fad referred to as the "new new Math."

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Extra resources for A Gardner's Workout: Training the Mind and Entertaining the Spirit

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Dens(F ) = 21 dens(F ) ❀ ❧♦♦s❡❧② s♣❡❛❦✐♥❣✱ t❤❡ ❝♦♥❞✐t✐♦♥ p ∈ PX ✐s ✐♥❞❡♣❡♥❞❡♥t ♦❢ ❛♥② ❢r♦❜❡♥✐❛♥ ❝♦♥❞✐t✐♦♥✳ ❈❤❛♣t❡r ✹ ❘❡✈✐❡✇ ♦❢ ✲❛❞✐❝ ❝♦❤♦♠♦❧♦❣② ❚❤❡ r❡s✉❧ts s✉♠♠❛r✐③❡❞ ✐♥ t❤✐s ❝❤❛♣t❡r ✭❡①❝❡♣t t❤♦s❡ ♦❢ ➓✹✳✻✮ ❝❛♥ ❜❡ ❢♦✉♥❞ ✐♥ t❤❡ t❤r❡❡ ✈♦❧✉♠❡s ♦❢ ❙●❆ r❡❧❛t✐✈❡ t♦ ét❛❧❡ ❝♦❤♦♠♦❧♦❣② ✿ ❬❙●❆ ✹❪✱ ❬❙●❆ 4 12 ❪ ❛♥❞ ❬❙●❆ ✺❪✱ t♦❣❡t❤❡r ✇✐t❤ ❉❡❧✐❣♥❡✬s ♣❛♣❡rs ❬❉❡ ✼✹❪ ❛♥❞ ❬❉❡ ✽✵❪ ♦♥ ❲❡✐❧✬s ❝♦♥❥❡❝t✉r❡s✳ ❋♦r ❛ s❤♦rt❡r ❛❝❝♦✉♥t ✭✇✐t❤ ♦r ✇✐t❤♦✉t ♣r♦♦❢s✮✱ s❡❡ ❡✳❣✳ ❬❋❑ ✽✽❪✱ ❬❑❛ ✾✹❪✱ ❬❑❛ ✵✶❜❪ ♦r ❬▼✐ ✽✵❪✳ ✹✳✶✳ ❚❤❡ ✲❛❞✐❝ ❝♦❤♦♠♦❧♦❣② ❣r♦✉♣s ks ❛♥❞ ❧❡t Γk = Gal(ks /k).

R❡ ❛r❡ ♥❛t✉r❛❧ ♠❛♣s ✭❞✉❡ t♦ t❤❡ ❢❛❝t t❤❛t t❤❡ ✉s✉❛❧ t♦♣♦❧♦❣② ✐s ✜♥❡r t❤❛♥ t❤❡ ét❛❧❡ ♦♥❡✮ ✿ H i (X, Q ) → H i (X(C), Q) ⊗ Q ❛♥❞ Hci (X, Q ) → Hci (X(C), Q) ⊗ Q . , xqn ). k ✲♣♦✐♥t x ♦❢ X ✐s k ✲r❛t✐♦♥❛❧ ✐❢ ❛♥❞ ♦♥❧② ✐❢ ✐t ✐s ✜①❡❞ ✉♥❞❡r F ✳ ❙✐♠✐❧❛r❧②✱ ✐❢ m ✐s ❛♥ ✐♥t❡❣❡r > 0✱ ❛♥❞ ✐❢ km ❞❡♥♦t❡s t❤❡ s✉❜❡①t❡♥s✐♦♥ ♦❢ k ♦❢ ❞❡❣r❡❡ m ♦✈❡r k ✱ t❤❡♥ X(km ) ✐s t❤❡ s✉❜s❡t m ♦❢ X(k) ♠❛❞❡ ✉♣ ♦❢ t❤❡ ♣♦✐♥ts ✜①❡❞ ✉♥❞❡r t❤❡ m✲t❤ ✐t❡r❛t❡ F ♦❢ F ✳ ❚❤❡ ♠♦r♣❤✐s♠ F : X → X ✐s ♣r♦♣❡r ❀ ❤❡♥❝❡ ✐t ❛❝ts ❜② ❢✉♥❝t♦r✐❛❧✐t② ♦♥ i t❤❡ ❝♦❤♦♠♦❧♦❣② s♣❛❝❡s Hc (X, Q )✱ ✇❤❡r❡ ✐s ❛♥② ♣r✐♠❡ ♥✉♠❜❡r = p✳ ▲❡t ✉s ❞❡♥♦t❡ ❜② Tri (F ) t❤❡ tr❛❝❡ ♦❢ t❤✐s ❡♥❞♦♠♦r♣❤✐s♠✱ ❛♥❞ ❞❡✜♥❡ ✿ ❖♥❡ ♦❢ ✐ts ♠❛✐♥ ♣r♦♣❡rt✐❡s ✐s t❤❛t ❛ (−1)i Tri (F ).

Tr(F ) = i ❚❤✐s ✐s t❤❡ ▲❡❢s❝❤❡t③ ♥✉♠❜❡r ♦❢ F✱ r❡❧❛t✐✈❡ t♦ t❤❡ ✲❛❞✐❝ ❝♦❤♦♠♦❧♦❣② ✇✐t❤ ♣r♦♣❡r s✉♣♣♦rt✳ ❆ ♣r✐♦r✐✱ ✐t ❞❡♣❡♥❞s ♦♥ t❤❡ ❝❤♦✐❝❡ ♦❢ ✳ ■♥ ❢❛❝t✱ ✐t ❞♦❡s ♥♦t✱ ❜❡❝❛✉s❡ ♦❢ t❤❡ ❢♦❧❧♦✇✐♥❣ r❡s✉❧t ♦❢ ●r♦t❤❡♥❞✐❡❝❦ ✭❬●r ✻✹❪✱ s❡❡ ❛❧s♦ ❬❙●❆ 4 21 ✱ ♣✳✽✻✱ t❤✳✸✳✷❪✮ ✿ ❚❤❡♦r❡♠ ✹✳✷✳ Tr(F ) = |X(k)|. ❚❤✐s ❛❧s♦ ❛♣♣❧✐❡s t♦ t❤❡ ✜♥✐t❡ ❡①t❡♥s✐♦♥s ♦❢ ❈♦r♦❧❧❛r② ✹✳✸✳ Tr(F m ) = |X(km )| ❢♦r ❡✈❡r② k✳ ❍❡♥❝❡ ✿ m 1✳ ❘❡♠❛r❦s✳ ✶✮ ❙✐♥❝❡ F :X→X ✐s ❛ r❛❞✐❝✐❛❧ ♠♦r♣❤✐s♠✱ ✐t ✐s ❛♥ ❤♦♠❡♦♠♦r♣❤✐s♠ ❢♦r t❤❡ ét❛❧❡ t♦♣♦❧♦❣②✳ ❍❡♥❝❡ ❡✈❡r② ❡✐❣❡♥✈❛❧✉❡ ♦❢ F ♦♥ Hci (X, Q ) ✐s ♥♦♥✲③❡r♦ ❀ ❢♦r ❛ ♠♦r❡ ♣r❡❝✐s❡ st❛t❡♠❡♥t✱ s❡❡ ❚❤❡♦r❡♠ ✹✳✺ ❜❡❧♦✇✳ ✷✮ ❚❤❡ t❤❡♦r❡♠ ♣r♦✈❡❞ ❜② ●r♦t❤❡♥❞✐❡❝❦✱ ❧♦❝✳❝✐t✳✱ ✐s ♠♦r❡ ❣❡♥❡r❛❧ t❤❛♥ ❚❤❡♦r❡♠ ✹✳✷ ✿ ✐t ❛♣♣❧✐❡s t♦ ❡✈❡r② ❝♦♥str✉❝t✐❜❧❡ ❛s ❛ s✉♠ ♦❢ ❧♦❝❛❧ tr❛❝❡s ❛t t❤❡ ♣♦✐♥ts ♦❢ ✸✮ ❆ss✉♠❡ k = Fp ✱ Q ✲s❤❡❛❢✱ ❛♥❞ ❣✐✈❡s Tr(F ) X(k)✳ t♦ s✐♠♣❧✐❢② ♥♦t❛t✐♦♥s✳ ❚❤❡♥ ❈♦r♦❧❧❛r② ✹✳✸ ✐s ❡q✉✐✈❛✲ ❧❡♥t t♦ s❛②✐♥❣ t❤❛t t❤❡ ❉✐r✐❝❤❧❡t s❡r✐❡s ❞❡♥♦t❡❞ ❜② ζX,p (s) ✐♥ ➓✶✳✺ ✐s ❡q✉❛❧ ✸✹ ✹✳ ❘❡✈✐❡✇ ♦❢ −s F |Hci (X, Q i det(1 − p ▼♦r❡♦✈❡r✱ ♦♥❡ ❤❛s t♦ i+1 ))(−1) NX (pe ) = ✲❛❞✐❝ ❝♦❤♦♠♦❧♦❣② ✱ ✇❤✐❝❤ ✐s ❛ r❛t✐♦♥❛❧ ❢✉♥❝t✐♦♥ ♦❢ p−s ✳ (−1)i Tri (F e ) i e ∈ Z ✭❛♥❞ ♥♦t ♠❡r❡❧② ❢♦r e 1✮✳ ■♥ ♣❛rt✐❝✉❧❛r✱ NX (p0 ) ✐s ❡q✉❛❧ t♦ i i (−1) dim Hc (X, Q )✱ ✇❤✐❝❤ ✐s t❤❡ ❊✉❧❡r✲P♦✐♥❝❛ré ❝❤❛r❛❝t❡r✐st✐❝ ♦❢ X ✳ ❢♦r ❡✈❡r② i ✹✳✹✳ ❚❤❡ ❝❛s❡ ♦❢ ❛ ✜♥✐t❡ ✜❡❧❞ ✿ t❤❡ ❣❡♦♠❡tr✐❝ ❛♥❞ t❤❡ ❛r✐t❤♠❡t✐❝ ❋r♦❜❡♥✐✉s ❑❡❡♣ t❤❡ ♥♦t❛t✐♦♥ ♦❢ ➓✹✳✸✳ ❚❤❡ ●❛❧♦✐s ❣r♦✉♣ i ❝♦❤♦♠♦❧♦❣② ❣r♦✉♣ Hc (X, Q ♦❢ Γk )✳ Γk = Gal(k/k) ❛❝ts ♦♥ ❡❛❝❤ σ = σq σ ✮✱ t❤❛t ✐s ❝❛❧❧❡❞ t❤❡ ❛r✐t❤✲ ■♥ ♣❛rt✐❝✉❧❛r✱ t❤❡ ♥❛t✉r❛❧ ❣❡♥❡r❛t♦r ❛❝ts ❜② ❛♥ ❛✉t♦♠♦r♣❤✐s♠ ✭st✐❧❧ ❞❡♥♦t❡❞ ❜② ♠❡t✐❝ ❋r♦❜❡♥✐✉s ❛✉t♦♠♦r♣❤✐s♠ ✐♥ ♦r❞❡r t♦ ❞✐st✐♥❣✉✐s❤ ✐t ❢r♦♠ t❤❡ ❣❡♦♠❡tr✐❝ ❋r♦❜❡♥✐✉s F ❞❡✜♥❡❞ ❛❜♦✈❡✳ ❚❤❡s❡ t✇♦ ❦✐♥❞ ♦❢ ✏❋r♦❜❡♥✐✉s ❛✉t♦♠♦r♣❤✐s♠s✑ ❛r❡ r❡❧❛t❡❞ ❜② t❤❡ ❢♦❧❧♦✇✐♥❣ s✐♠♣❧❡ r❡s✉❧t ✭s❡❡ ❬❙●❆ ✺✱ ♣✳✹✺✼❪✱ ♦r ❬❑❛ ✾✹✱ ✷✹✲✷✺❪✮ ✿ ❚❤❡♦r❡♠ ✹✳✹✳ ❚❤❡ ❛r✐t❤♠❡t✐❝ ❋r♦❜❡♥✐✉s ❛♥❞ t❤❡ ❣❡♦♠❡tr✐❝ ❋r♦❜❡♥✐✉s ❛r❡ ✐♥✈❡rs❡s ♦❢ ❡❛❝❤ ♦t❤❡r✳ ■♥ ♦t❤❡r ✇♦r❞s✱ ✇❡ ❤❛✈❡ σ(F ξ) = F (σξ) = ξ ❢♦r ❡✈❡r② ❆ s✐♠✐❧❛r r❡s✉❧t ❤♦❧❞s ❢♦r t❤❡ ❝♦❤♦♠♦❧♦❣② ❣r♦✉♣s ξ ∈ Hci (X, Q )✳ H i (X, Q )✱ ❜✐tr❛r② s✉♣♣♦rt ✭❛♥❞ ❛❧s♦ ❢♦r t❤❡ ❝♦❤♦♠♦❧♦❣② ✇✐t❤ ❝♦❡✣❝✐❡♥ts ✐♥ ❊①❛♠♣❧❡✳ ❙✉♣♣♦s❡ t❤❛t ❚❛t❡ Q X ✐s ❛♥ ❛❜❡❧✐❛♥ ✈❛r✐❡t② ♦✈❡r ✇✐t❤ ❛r✲ Z/ n Z✮✳ k ✱ ❛♥❞ ❧❡t V (X) ❜❡ ✐ts ✲♠♦❞✉❧❡✳ ❬❘❡❝❛❧❧ t❤❛t V (X) = Q ⊗ lim X[ n ]✱ ✇❤❡r❡ X[ n ] ✐s t❤❡ ❣r♦✉♣ ♦❢ t❤❡ n ✲❞✐✈✐s✐♦♥ ←− ♣♦✐♥ts ♦❢ X(k)✱ ✐✳❡✳ t❤❡ ❦❡r♥❡❧ ♦❢ n : X(k) → X(k✮ ❀ ✐t ✐s ❛ Q ✲✈❡❝t♦r s♣❛❝❡ ♦❢ ❞✐♠❡♥s✐♦♥ 2dim X ✳❪ ❚❤❡ ❋r♦❜❡♥✐✉s ❡♥❞♦♠♦r♣❤✐s♠ ♠❡t✐❝ ❋r♦❜❡♥✐✉s F s F : X → X ❛❝ts ♦♥ V (X) ❀ t❤❡ ❛r✐t❤✲ ❛❧s♦ ❛❝ts✱ ❛♥❞ ✐ts ❛❝t✐♦♥ ✐s t❤❡ s❛♠❡ ❛s t❤❡ ❛❝t✐♦♥ ♦❢ F ❛♥❞ s ❛❝t ✐♥ t❤❡ s❛♠❡ ✇❛② ♦♥ X(k)✮✳ ❚❤❡ ✜rst ❝♦❤♦♠♦❧♦❣② H 1 (X, Q ) ✐s t❤❡ ❞✉❛❧ ♦❢ V (X) ❀ t❤❡ ❛❝t✐♦♥ ♦❢ F ♦♥ ✐t ✐s ❞❡✜♥❡❞ ❜② ❢✉♥❝t♦r✐❛❧✐t②✱ ✐✳❡✳ ❜② tr❛♥s♣♦s✐t✐♦♥ ❀ t❤❡ ❛❝t✐♦♥ ♦❢ s ✐s ❞❡✜♥❡❞ ❜② tr❛♥s♣♦rt ♦❢ ✭❜❡❝❛✉s❡ ❣r♦✉♣ str✉❝t✉r❡✱ ✐✳❡✳ ❜② ✐♥✈❡rs❡ tr❛♥s♣♦s✐t✐♦♥✳ ❚❤✐s ❡①♣❧❛✐♥s ✇❤② t❤❡ t✇♦ ❛❝t✐♦♥s ❛r❡ ✐♥✈❡rs❡ ♦❢ ❡❛❝❤ ♦t❤❡r✳ ✸ ❲❤❛t ✸ t❤✐s ❡①❛♠♣❧❡ s✉❣❣❡sts ✐s t❤❛t✱ ✐❢ ét❛❧❡ t♦♣♦❧♦❣② ✇❡r❡ ❡①♣r❡ss❡❞ ✐♥ t❡r♠s ♦❢ ❤♦♠♦❧♦❣② ✐♥st❡❛❞ ♦❢ ❝♦❤♦♠♦❧♦❣②✱ t❤❡ t✇♦ t②♣❡s ♦❢ ❋r♦❜❡♥✐✉s ✇♦✉❧❞ ❜❡ t❤❡ s❛♠❡✳ ✹✳✺✳ ✳ ❚❤❡ ❝❛s❡ ♦❢ ❛ ✜♥✐t❡ ✜❡❧❞ ✿ ❉❡❧✐❣♥❡✬s t❤❡♦r❡♠s ✸✺ ✹✳✺✳ ❚❤❡ ❝❛s❡ ♦❢ ❛ ✜♥✐t❡ ✜❡❧❞ ✿ ❉❡❧✐❣♥❡✬s t❤❡♦r❡♠s ❲❡ ❦❡❡♣ t❤❡ ♥♦t❛t✐♦♥ ❛♥❞ ❤②♣♦t❤❡s❡s ♦❢ ➓✹✳✹ ❛❜♦✈❡✳ q ✲❲❡✐❧ ✐♥t❡❣❡r ♦❢ ✇❡✐❣❤t w ∈ N ✐s ❛♥ ❛❧❣❡❜r❛✐❝ ✐♥t❡❣❡r α |ι(α)| = q w/2 ❢♦r ❡✈❡r② ❡♠❜❡❞❞✐♥❣ ι : Q(α) → C✳ ❋♦r ✐♥st❛♥❝❡ ❛ ❘❡❝❛❧❧ t❤❛t ❛ s✉❝❤ t❤❛t q ✲❲❡✐❧ ✐♥t❡❣❡r ♦❢ ✇❡✐❣❤t ✵ ✐s ❛ r♦♦t ♦❢ ✉♥✐t② ✭❑r♦♥❡❝❦❡r✮✳ q ✲❲❡✐❧ ✐♥t❡❣❡r ♦❢ ✇❡✐❣❤t ✇❡✐❣❤t w r❡❧❛t✐✈❡❧② t♦ q ✑✳ ❘❡♠❛r❦✳ ■♥ ❉❡❧✐❣♥❡ ❬❉❡ ✽✵✱ ➓✶✳✷✳✶❪✱ ✇❤❛t ✇❡ ❝❛❧❧ ❛ w ✐s ❝❛❧❧❡❞ ✏ ❛♥ ❛❧❣❡❜r❛✐❝ ✐♥t❡❣❡r t❤❛t ✐s ♣✉r❡ ♦❢ ❚❤❡♦r❡♠ ✹✳✺✳ ✭❉❡❧✐❣♥❡✮ ▲❡t d = dim X ✳ α ♦❢ t❤❡ ❣❡♦♠❡tr✐❝ ❋r♦❜❡♥✐✉s F ❛❝t✐♥❣ ♦♥ Hci (X, Q ) ✐s ❛ q ✲❲❡✐❧ ✐♥t❡❣❡r ♦❢ ✇❡✐❣❤t i ; ✐❢ i d✱ t❤❡♥ α ✐s ❞✐✈✐s✐❜❧❡ ❜② q i−d .

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