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❚♦rr❡s ❞❡ ❍❛♥♦✐✿ ❡♣í❧♦❣♦ ❋♦♥t❡✿ ❤tt♣✿✴✴❡♥✳✇✐❦✐♣❡❞✐❛✳♦r❣✴

❚♦rr❡s ❞❡ ❍❛♥♦✐ ❆ ❇ ❈ ❉❡s❡❥❛♠♦s tr❛♥s❢❡r✐r ♥ ❞✐s❝♦s ❞♦ ♣✐♥♦ ❆ ♣❛r❛ ♦ ♣✐♥♦ ❈ ✉s❛♥❞♦ ♦ ♣✐♥♦ ❇ ❝♦♠♦ ❛✉①✐❧✐❛r ❡ r❡♣❡✐t❛♥❞♦ ❛s r❡❣r❛s✿ ◮ ♣♦❞❡♠♦s ♠♦✈❡r ❛♣❡♥❛s ✉♠ ❞✐s❝♦ ♣♦r ✈❡③❀ ◮ ♥✉♥❝❛ ✉♠ ❞✐s❝♦ ❞❡ ❞✐â♠❡tr♦ ♠❛✐♦r ♣♦❞❡rá s❡r ❝♦❧♦❝❛❞♦ s♦❜r❡ ✉♠ ❞✐s❝♦ ❞❡ ❞✐â♠❡tr♦ ♠❡♥♦r✳

✶✳ r❡s♦❧✈❡r ❍❛♥♦✐✭ ♥ ✲✶✱ ❆ ✱ ❈ ✱ ❇ ✮ ✷✳ ♠♦✈❡r ♦ ❞✐s❝♦ ♥ ❞❡ ❆ ♣❛r❛ ❈ ✸✳ r❡s♦❧✈❡r ❍❛♥♦✐✭ ♥ ✲✶✱ ❇ ✱ ❆ ✱ ❈ ✮ ❇❛s❡ ✿ s❛❜❡♠♦s r❡✈♦❧✈❡r ❍❛♥♦✐✭ ✵ ✱✳ ✳ ✳ ✱✳ ✳ ✳ ✱✳ ✳ ✳ ✮ ❆❧❣♦r✐t♠♦ r❡❝✉rs✐✈♦ ❆ ❇ ❈ P❛r❛ r❡s♦❧✈❡r ❍❛♥♦✐✭ ♥ ✱ ❆ ✱ ❇ ✱ ❈ ✮ ❜❛st❛✿

✷✳ ♠♦✈❡r ♦ ❞✐s❝♦ ♥ ❞❡ ❆ ♣❛r❛ ❈ ✸✳ r❡s♦❧✈❡r ❍❛♥♦✐✭ ♥ ✲✶✱ ❇ ✱ ❆ ✱ ❈ ✮ ❇❛s❡ ✿ s❛❜❡♠♦s r❡✈♦❧✈❡r ❍❛♥♦✐✭ ✵ ✱✳ ✳ ✳ ✱✳ ✳ ✳ ✱✳ ✳ ✳ ✮ ❆❧❣♦r✐t♠♦ r❡❝✉rs✐✈♦ ❆ ❇ ❈ P❛r❛ r❡s♦❧✈❡r ❍❛♥♦✐✭ ♥ ✱ ❆ ✱ ❇ ✱ ❈ ✮ ❜❛st❛✿ ✶✳ r❡s♦❧✈❡r ❍❛♥♦✐✭ ♥ ✲✶✱ ❆ ✱ ❈ ✱ ❇ ✮

✸✳ r❡s♦❧✈❡r ❍❛♥♦✐✭ ♥ ✲✶✱ ❇ ✱ ❆ ✱ ❈ ✮ ❇❛s❡ ✿ s❛❜❡♠♦s r❡✈♦❧✈❡r ❍❛♥♦✐✭ ✵ ✱✳ ✳ ✳ ✱✳ ✳ ✳ ✱✳ ✳ ✳ ✮ ❆❧❣♦r✐t♠♦ r❡❝✉rs✐✈♦ . ❆ ❇ ❈ P❛r❛ r❡s♦❧✈❡r ❍❛♥♦✐✭ ♥ ✱ ❆ ✱ ❇ ✱ ❈ ✮ ❜❛st❛✿ ✶✳ r❡s♦❧✈❡r ❍❛♥♦✐✭ ♥ ✲✶✱ ❆ ✱ ❈ ✱ ❇ ✮ ✷✳ ♠♦✈❡r ♦ ❞✐s❝♦ ♥ ❞❡ ❆ ♣❛r❛ ❈

❆❧❣♦r✐t♠♦ r❡❝✉rs✐✈♦ . ❆ ❇ ❈ P❛r❛ r❡s♦❧✈❡r ❍❛♥♦✐✭ ♥ ✱ ❆ ✱ ❇ ✱ ❈ ✮ ❜❛st❛✿ ✶✳ r❡s♦❧✈❡r ❍❛♥♦✐✭ ♥ ✲✶✱ ❆ ✱ ❈ ✱ ❇ ✮ ✷✳ ♠♦✈❡r ♦ ❞✐s❝♦ ♥ ❞❡ ❆ ♣❛r❛ ❈ ✸✳ r❡s♦❧✈❡r ❍❛♥♦✐✭ ♥ ✲✶✱ ❇ ✱ ❆ ✱ ❈ ✮ ❇❛s❡ ✿ s❛❜❡♠♦s r❡✈♦❧✈❡r ❍❛♥♦✐✭ ✵ ✱✳ ✳ ✳ ✱✳ ✳ ✳ ✱✳ ✳ ✳ ✮

❚❤❡ ❚♦✇❡r ♦❢ ❍❛♥♦✐ ❙t♦r② Taken From W.W. Rouse Ball & H.S.M. Coxeter, Mathematical Recreations and Essays, 12th edition. Univ. of Toronto Press, 1974. The De Parville account of the origen from La Nature, Paris, 1884, part I, pp. 285-286. ■♥ t❤❡ ❣r❡❛t t❡♠♣❧❡ ❛t ❇❡♥❛r❡s ❜❡♥❡❛t❤ t❤❡ ❞♦♠❡ t❤❛t ♠❛r❦s t❤❡ ❝❡♥tr❡ ♦❢ t❤❡ ✇♦r❧❞✱ r❡sts ❛ ❜r❛ss ♣❧❛t❡ ✐♥ ✇❤✐❝❤ ❛r❡ ✜①❡❞ t❤r❡❡ ❞✐❛♠♦♥❞ ♥❡❡❞❧❡s✱ ❡❛❝❤ ❛ ❝✉❜✐t ❤✐❣❤ ❛♥❞ ❛s t❤✐❝❦ ❛s t❤❡ ❜♦❞② ♦❢ ❛ ❜❡❡✳ ❖♥ ♦♥❡ ♦❢ t❤❡s❡ ♥❡❡❞❧❡s✱ ❛t t❤❡ ❝r❡❛t✐♦♥✱ ●♦❞ ♣❧❛❝❡❞ s✐①t②✲❢♦✉r ❞✐s❝s ♦❢ ♣✉r❡ ❣♦❧❞✱ t❤❡ ❧❛r❣❡st ❞✐s❦ r❡st✐♥❣ ♦♥ t❤❡ ❜r❛ss ♣❧❛t❡✱ ❛♥❞ t❤❡ ♦t❤❡rs ❣❡tt✐♥❣ s♠❛❧❧❡r ❛♥❞ s♠❛❧❧❡r ✉♣ t♦ t❤❡ t♦♣ ♦♥❡✳ ❚❤✐s ✐s t❤❡ t♦✇❡r ♦❢ ❇r❛♠❛❤✳ ❉❛② ❛♥❞ ♥✐❣❤t ✉♥❝❡❛s✐♥❣❧② t❤❡ ♣r✐❡st tr❛♥s❢❡r t❤❡ ❞✐s❝s ❢r♦♠ ♦♥❡ ❞✐❛♠♦♥❞ ♥❡❡❞❧❡ t♦ ❛♥♦t❤❡r ❛❝❝♦r❞✐♥❣ t♦ t❤❡ ✜①❡❞ ❛♥❞ ✐♠♠✉t❛❜❧❡ ❧❛✇s ♦❢ ❇r❛♠❛❤✱ ✇❤✐❝❤ r❡q✉✐r❡ t❤❛t t❤❡ ♣r✐❡st ♦♥ ❞✉t② ♠✉st ♥♦t ♠♦✈❡ ♠♦r❡ t❤❛♥ ♦♥❡ ❞✐s❝ ❛t ❛ t✐♠❡ ❛♥❞ t❤❛t ❤❡ ♠✉st ♣❧❛❝❡ t❤✐s ❞✐s❝ ♦♥ ❛ ♥❡❡❞❧❡ s♦ t❤❛t t❤❡r❡ ✐s ♥♦ s♠❛❧❧❡r ❞✐s❝ ❜❡❧♦✇ ✐t✳ ❲❤❡♥ t❤❡ s✐①t②✲❢♦✉r ❞✐s❝s s❤❛❧❧ ❤❛✈❡ ❜❡❡♥ t❤✉s tr❛♥s❢❡rr❡❞ ❢r♦♠ t❤❡ ♥❡❡❞❧❡ ✇❤✐❝❤ ❛t ❝r❡❛t✐♦♥ ●♦❞ ♣❧❛❝❡❞ t❤❡♠✱ t♦ ♦♥❡ ♦❢ t❤❡ ♦t❤❡r ♥❡❡❞❧❡s✱ t♦✇❡r✱ t❡♠♣❧❡✱ ❛♥❞ ❇r❛❤♠✐♥s ❛❧✐❦❡ ✇✐❧❧ ❝r✉♠❜❧❡ ✐♥t♦ ❞✉st ❛♥❞ ✇✐t❤ ❛ t❤✉♥❞❡r❝❧❛♣ t❤❡ ✇♦r❧❞ ✇✐❧❧ ✈❛♥✐s❤✳ ❚❤❡ ♥✉♠❜❡r ♦❢ s❡♣❛r❛t❡ tr❛♥s❢❡rs ♦❢ s✐♥❣❧❡ ❞✐s❝s ✇❤✐❝❤ t❤❡ ❇r❛❤♠✐♥s ♠✉st ♠❛❦❡ t♦ ❡✛❡❝t t❤❡ tr❛♥s❢❡r ♦❢ t❤❡ t♦✇❡r ✐s t✇♦ r❛✐s❡❞ t♦ t❤❡ s✐①t②✲❢♦✉rt❤ ♣♦✇❡r ♠✐♥✉s ✶ ♦r ✶✽✱✹✹✻✱✼✹✹✱✵✼✸✱✼✵✾✱✺✺✶✱✻✶✺ ♠♦✈❡s✳ ❊✈❡♥ ✐❢ t❤❡ ♣r✐❡sts ♠♦✈❡ ♦♥❡ ❞✐s❦ ❡✈❡r② s❡❝♦♥❞✱ ✐t ✇♦✉❧❞ t❛❦❡ ♠♦r❡ t❤❛♥ ✺✵✵ ❜✐❧❧✐♦♥ ②❡❛rs t♦ r❡❧♦❝❛t❡ t❤❡ ✐♥✐t✐❛❧ t♦✇❡r ♦❢ ✻✹ ❞✐s❦s✳ ❤tt♣✿✴✴✇✇✇✳r❝✐✳r✉t❣❡rs✳❡❞✉✴✄❝❢s✴✹✼✷❴❤t♠❧✴❆■❴❙❊❆❘❈❍✴❙t♦r②❴❚❖❍✳❤t♠❧

◆ú♠❡r♦ ❞❡ ♠♦✈✐♠❡♥t♦s ❙❡❥❛ ❚ ✭ ♥ ✮ ♦ ♥ú♠❡r♦ ❞❡ ♠♦✈✐♠❡♥t♦s ❢❡✐t♦s ♣❡❧♦ ❛❧❣♦r✐t♠♦ ♣❛r❛ r❡s♦❧✈❡r ♦ ♣r♦❜❧❡♠❛ ❞❛s t♦rr❡s ❞❡ ❍❛♥♦✐ ❝♦♠ ♥ ❞✐s❝♦✳ ❚❡♠♦s q✉❡ ❚ ( 0 ) = 0 ❚ ( n ) = 2 ❚ ( ♥ − 1) + 1 ♣❛r❛ n = 1 , 2 , 3 . . . ◗✉❛♥t♦ ✈❛❧❡ ❚ ✭ ♥ ✮❄

❘❡❝♦rrê♥❝✐❛ ❚❡♠♦s q✉❡ ❚ ( ♥ ) = 2 ❚ ( ♥ − 1) + 1 = 2 (2 ❚ ( ♥ − 2) + 1) + 1 = 2 (2 (2 ❚ ( ♥ − 3) + 1) + 1) + 1 = 2 (2 (2 (2 ❚ ( ♥ − 4) + 1) + 1) + 1) + 1 = · · · = 2 (2 (2 (2 ( · · · (2 ❚ (0) + 1)) + 1) + 1) + 1

❘❡❝♦rrê♥❝✐❛ ▲♦❣♦✱ ❚ ( ♥ ) = 2 n − 1 + · · · + 2 3 + 2 2 + 2 + 1 = 2 n − 1 . ♥ 0 1 2 3 4 5 6 7 8 9 ❚ ( ♥ ) 0 1 3 7 15 31 63 127 255 511

❈♦♥❝❧✉sõ❡s ❖ ♥ú♠❡r♦ ❞❡ ♠♦✈✐♠❡♥t♦s ❢❡✐t♦s ♣❡❧❛ ❝❤❛♠❛❞❛ ❤❛♥♦✐✭♥✱ ✳✳✳✱ ✳✳✳✱ ✳✳✳✮ é 2 n − 1 . ◆♦t❡♠♦s q✉❡ ❛ ❢✉♥çã♦ ❤❛♥♦✐ ❢❛③ ♦ ♥ú♠❡r♦ ♠í♥✐♠♦ ❞❡ ♠♦✈✐♠❡♥t♦s✿ ♥ã♦ é ♣♦ssí✈❡❧ r❡s♦❧✈❡r ♦ q✉❡❜r❛✲❝❛❜❡ç❛ ❝♦♠ ♠❡♥♦s ♠♦✈✐♠❡♥t♦s✳

❊♥q✉❛♥t♦ ✐st♦ ✳ ✳ ✳ ♦s ♠♦♥❣❡s ✳ ✳ ✳ ❚ (64) = 18 . 446 . 744 . 073 . 709 . 551 . 615 ≈ 1 , 84 × 10 19 ❙✉♣♦♥❤❛ q✉❡ ♦s ♠♦♥❣❡s ❢❛ç❛♠ ♦ ♠♦✈✐♠❡♥t♦ ❞❡ ✶ ❞✐s❝♦ ♣♦r s❡❣✉♥❞♦✭✦✮✳ 1 , 8 × 10 19 s❡❣ ≈ 3 , 07 × 10 17 ♠✐♥ ≈ 5 , 11 × 10 15 ❤♦r❛s ≈ 2 , 13 × 10 14 ❞✐❛s ≈ 5 , 83 × 10 11 ❛♥♦s . = 583 ❜✐❧❤õ❡s ❞❡ ❛♥♦s . ❆ ✐❞❛❞❡ ❞❛ ❚❡rr❛ é ✹✱✺✹ ❜✐❧❤õ❡s ❞❡ ❛♥♦s ✳