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ÈÂappleÓ· ‰ÎÊ ±±∂ ÔÂÈÏÈ‚· Ìȯ˙ÂÙ‰ ÔÈ·Ó<br />
ÆÚ·˘ ¯‡·Ó ̘ χÎÈÓ ¢Ï·‰ ͯ„¢Ï<br />
ÔÂÁ¯ÈÏ ÈÂappleÓ Ï¯‚ÂÈ Ìȯ˙ÂÙ‰ ÔÈ·<br />
ÂÏÎÂ˙ ÔÂÁ¯È· ƢϷ‰ ͯ„¢<br />
¨ÌȇÏÙapple ÌÈappleÂÎ˙Ó ‡ÂˆÓÏ<br />
¨˙ˆÚ ÌÈÙÈË ¨˙Â„Ú ÈÏ·Ó<br />
¨ÌÈÏÈ·ÂÓ ÌÈÙ˘ ÌÚ ˙ÂappleÂȇȯ<br />
Æ„ÂڠωÂÎχ ¨ÔÂÊÓ ÈËÂ˘È˜<br />
ÁÂÏ˘Ï ˘È ˙Âapple¯˙Ù‰ ˙‡<br />
¨¢Â‡ÈÏÈÏ‚ ˙ίÚÓ¢Ï<br />
¨¯Â‡Ï ‰‡ˆÂ‰ SBC<br />
∏ ÏË ÚÙ˘ ßÁ¯<br />
∂∑∞±≥ ¨·È·‡≠Ï˙<br />
shlomit@sbc.co.il ∫Ï Â‡<br />
‡˘Âapple‰ ˙¯Â˘· ÔÈȈÏÂ<br />
Æ¢˘‡¯ ˙ˆÈÙ˜ ¯Â„ÓÏ¢<br />
ÆÔÂÙÏË ˙·Â˙Î ¨‡ÏÓ Ì˘ ÔÈÈˆÏ ‡apple<br />
˙Âχ˘‰Ó ˜ÏÁÏ ˙Âapple¯˙Ù Ì‚<br />
Ɖϯ‚‰· ÂÙ˙˙˘È „·Ï·<br />
˙Ú„Ï ÔÈÈappleÚÓ Æ‰ÏȘ˘‰ ˙„ÈÁ ÌÂÁ˙· ¯˙ÂÈ· ˙¢˜‰ ˙ÂÈÚ·Ï ˙·˘Áapple ԇΠ‚ˆÂ‰˘ ˙ÂÈÚ·‰<br />
Ô˘‰ ˙ÂÏȘ˘‰ ¯ÙÒÓ· ‰Ï‡Î ˙ÂÈÚ· Ô¯˙ÙÏ ˙ÂÈÏÏÎ ˙ÂËÈ˘ ÌÂȉ „Ú ÌÈ˘ÙÁÓ Ìȇ˜ÈËÓ˙Ó˘<br />
ÆÍη ÂÁÈψ‰ ‡Ï ÔÈÈ„Ú Ì‰ ͇ ¨¯˙ÂÈ·<br />
¯˘Ù‡˘ ˙ÂÚ·ËÓ Ï˘ ¯˙ÂÈ· Ï„‚‰ ¯ÙÒÓ‰ Â‰Ó ‡ÂˆÓÏ È„Î ·˘ÁÓ ˙ÂÈappleÎ˙· ÌȯÊÚapple ÌÈ·¯ Ìȯ˜ÂÁ<br />
ÌÈ˘Ó˙˘Ó ̇ ÆÛÈÂÊÓ Ú·ËÓ ˙‡„· ‡ˆÓapple˘ ÍÎ ˙ÂÙÎ ÈappleÊ‡Ó ÏÚ ˙ÂÏȘ˘ n ˙¯ÊÚ· Ϙ˘Ï<br />
Ì˙ȯ‚χ· ÌÈ˘Ó˙˘Ó ̇ Æ≥ n≠± ‡Â‰ ˙ÂÚ·ËÓ‰ ¯ÙÒÓ ¨¯Â„Ó‰ ˙ÏÈÁ˙· ‚ˆÂÓ‰ ËÂ˘Ù‰ Ì˙ȯ‚χ·<br />
Æ®‰ËÓÏ ‰¯Ú‰ ‡¯© ®≥ n ≠≥©Ø≤ ∫‡Â‰ ˙ÂÚ·ËÓ‰ ¯ÙÒÓ Èʇ ¨Èapple˘‰<br />
ÔÙ‡· Û‚ Ï˘ ÂϘ˘Ó „„Óapple ̉·˘© ÌÈÏÈ‚¯ ÌÈÈappleʇӷ ÌÈ˘Ó˙˘Ó ‰ÏȘ˘ ˙ÂÈÚ· Ï˘ ÛÒÂapple ‚ÂÒ·<br />
∫ÔÏ‰Ï˘ ‰Ï‡˘· ÂÊ ‰ÈÚ· ÌÈ‚„apple Æ®ËÏÁÂÓ<br />
È˙ÈÓ‡ Ú·ËÓ˘ ÚÂ„È Æ˙ÂÚ·ËÓ ‰¯˘Ú ˘È ‰Ó¯Ú Ïη Æ˙ÂÚ·ËÓ Ï˘ ˙ÂÓ¯Ú ¯˘Ú ˙ÂappleÂ˙apple ∫‰Ï‡˘<br />
ÏΠ¨ÌÈÙÈÂÊÓ ˙ÂÚ·ËÓ Ï˘ ‡È‰ ˙ÂÓ¯Ú‰ ˙Á‡ ÆÌÈÓ¯‚ π Ï˜Â˘ ÛÈÂÊÓ Ú·ËÓ ¨ÌÈÓ¯‚ ±∞ Ï˜Â˘<br />
˙ÂÚ·ËÓ‰ Ï˘ ÂÊ ‡È‰ ‰Ó¯Ú ÂÊȇ ‰Ï‚apple ¨˙Á‡ ‰ÏȘ˘ ˙¯ÊÚ· ¨Íȇ ÆÌÈÈ˙ÈÓ‡ ˙ÂÚ·ËÓ Ï˘ Ô‰ ¯‡˘‰<br />
øÌÈÙÈÂÊÓ‰<br />
˙ÂÚ·ËÓ ‰˘ÂÏ˘ ¨‰ÈÈapple˘‰ ‰Ó¯Ú‰Ó ˙ÂÚ·ËÓ Èapple˘ ¨‰apple¢‡¯‰ ‰Ó¯Ú‰Ó „Á‡ Ú·ËÓ Á˜Èapple ∫Ô¯˙Ù<br />
‰Ï‡‰ ˙ÂÚ·ËÓ‰ ÏÎ ˙‡ Æ˙È¯È˘Ú‰ ‰Ó¯Ú‰Ó ˙ÂÚ·ËÓ ‰¯˘Ú „Ú ¨‰‡Ï‰ ÔΠ˙È˘ÈÏ˘‰ ‰Ó¯Ú‰Ó<br />
Æ®ø„ˆÈΩ ˙ÙÈÂÊÓ‰ ‡È‰ ‰Ó¯Ú ÂÊȇ Ú„apple Ï·˜˙‰˘ Ϙ˘Ó‰ ÈÙÏ ¨ÌÈÈappleʇӷ Ϙ˘apple<br />
¯ÙÒ‚ „ÂϘ ±∂≠‰ ‰‡Ó‰ Ô· È˙Ù¯ˆ‰ ȇapple„ÈÁ‰Â ȇ˜ÈËÓ˙Ó‰ ‚Ȉ‰ ‰ÏȘ˘ ˙ÂÈÚ· Ï˘ ÛÒÂapple ‚ÂÒ<br />
ÈÚ¢ڢ ˙„ÈÁ· ˜ÒÚ˘ ÌÈ·¯ ÌȯÙÒ ¯·ÈÁ ‰˘‡· Æ®Bachet de Méziriac© ˜‡È¯ÈÊÓ ‰„ ‰˘‡·<br />
·˙Îapple˘ ‰Ï‡ Ìȇ˘Âapple· ÌȯÙÒ‰ ·Â¯Ï ÒÈÒ· ÂÂȉ ¨Ì‚ÂÒÓ ÌÈapple¢‡¯‰ ÔÓ Âȉ˘ ¨‰Ï‡ ÌȯÙÒ Æ‰˜ÈËÓ˙Ó<br />
Ư˙ÂÈ ˙¯ÁÂ‡Ó ÌÈapple˘·<br />
˙Âapple¢ ˙ÂϘ˘Ó Ï˘ ¯˙ÂÈ· Ô˘‰ ¯ÙÒÓ‰ Â‰Ó ∫˙appleÈÈappleÚÓ ‰ÏȘ˘ ˙ÈÈÚ· ‰ÚÈÙÂÓ ÂȯÙÒÓ „Á‡·<br />
ø˙ÂÙÎ Èappleʇӷ ¨‚¢˜ ¥∞≠Ï ± ÔÈ· ‰˘ÏΠϘ˘Ó Ϙ˘Ï È„Î Íȯˆ˘<br />
≥≤≠ ‚¢˜ ±∂ ¨‚¢˜ ∏ ¨‚¢˜ ¥ ¨‚¢˜ ≤ ¨‚¢˜ ± ∫˙ÂϘ˘Ó‰ ˘˘· ˘Ó˙˘‰Ï ¯˘Ù‡ ÈÎ ¨‰‡¯apple Ô¢‡¯ Ë·Ó·<br />
Æ‚¢˜ ¥∞≠Ï ‚¢˜ ± ÔÈ· ÂϘ˘Ó˘ ıÙÁ ÏΠϘ˘Ï È„Î ¨‚¢˜<br />
ÔÊ‡Ï ÏÎÂapple ¨ÌÈÈappleʇӉ Ï˘ ˙Á‡ Ûη ‚¢˜ ±± ÂϘ˘Ó˘ ‰Ó„‡ ÈÁÂÙ˙ Ï˘ ˜˘ ÌÈ˘apple ̇ ∫‰Ó‚„Ï<br />
ÆÌÈÈappleʇӉ Ï˘ ˙¯Á‡‰ Ûη ÌÈ˘apple˘ ¨‚¢˜ ∏≠ ‚¢˜ ≤ ¨‚¢˜ ± ∫˙ÂϘ˘Ó‰ ˘ÂÏ˘ ˙¯ÊÚ· Â˙‡<br />
Æ®ÌÈȯ‡appleÈ· ÌȯÙÒÓ© ≤ Ï˘ ˙˜ÊÁ ÔÏÂÎ Ô‰ Âapple˘Ó˙˘‰ Ô‰·˘ ˙ÂϘ˘Ó‰<br />
ÌȯÙÒÓ‰ ÏÎ ˙‡ Ì˙¯ÊÚ· ¯ÂˆÈÏ ¯˘Ù‡˘ ‡È‰ ÌÈȯ‡appleÈ· ÌȯÙÒÓ Ï˘ ˙ÂappleÈÈappleÚÓ‰ ˙ÂappleÂÎ˙‰ ˙Á‡<br />
¨ÍÎÈÙÏ ¨˙Á‡ ÌÚÙÓ ¯˙ÂÈ ®È¯‡appleÈ·© ¯ÙÒÓ· ˘Ó˙˘‰Ï ÈÏ·Ó ¨¯Â·ÈÁ‰ ˙ÏÂÚÙ ˙ÂÚˆÓ‡· ÌÈÓÏ˘‰<br />
Ɖ„ÈÁ‰ Ô¯˙ÙÏ ˙ÂÓȇ˙Ó ¢˙Âȯ‡appleÈ·¢ ˙ÂϘ˘Ó<br />
ÌÈ˘Ï Âapple„Ù˜‰ ÂÈ˘ÎÚ „Ú ÆÔ˘ ¢˜È¯Ë¢· ¯ÊÚÈapple ̇ ‰˘‡· Ï˘ ‰ÈÚ·Ï Ô¯˙Ù‰ ˙‡ ¯Ù˘Ï ¯˘Ù‡ ¨ÌχÂ<br />
˙ÂϘ˘Ó‰ ˙‡ ÌÈ˘apple ̇ Ï·‡ ƯÁ‡‰ „ˆ· ÌÈÏ˜Â˘ Â˙‡˘ ıÙÁ‰ ˙‡Â ¨„·Ï· „Á‡ „ˆ· ˙ÂϘ˘Ó‰ ˙‡<br />
˙ÈËÓ˙Ó ‰appleÈÁ·Ó Æ„·Ï· ˙Âapple¢ ˙ÂϘ˘Ó Ú·¯‡ ÌÚ ‰ÈÚ·‰ ˙‡ ¯Â˙ÙÏ ÏÎÂapple ¨ÌÈÈappleʇӉ È„ˆ Èapple˘Ó<br />
ÌÎÏ Âapple¯‡˘‰ ÔÎÏ ¨ÛÈΉ ˙‡ ÌÎÏ Ò¯‰Ï ¨Ô·ÂÓÎ ¨‰ˆ¯apple ‡Ï °¯ÂÒÈÁ‰ ˙ÏÂÚÙ· Ì‚Â ¨¯Â·ÈÁ ˙ÏÂÚÙ· ˘Ó˙˘apple<br />
ÆÔÈ·È ÔȷӉ ¨ÌÈȯ‡apple¯Ë ÌȯÙÒÓ· ÌÈ˘Ó˙˘Ó ÌÚÙ‰˘ ÊÂÓ¯apple ˜¯ Æ͢Ӊ· ÌÎÓˆÚ· ÂÊ ‰ÈÚ· ¯Â˙ÙÏ<br />
ÌÂÒ¯Ù ÌÚ ¯·‚˙‰ ‡˘Âapple· ÔÈÈappleÚ‰ Ï·‡ ¨„Â‡Ó ˙˜È˙Ú Ô‰ ‰ÏȘ˘ ˙ÂÈÚ· ƉȯÂËÒȉ ˙ˆ˜ ¨ÌÂÈÒÏ<br />
‚‰ ÈappleÏÂÙ≠Ȅ‰ȉ ȇ˜ÈËÓ˙Ó‰ Ï˘ ¨®Mathematical Snapshots© ¢ÌÈÈËÓ˙Ó ÌȘʷ‰¢ ¯ÙÒ‰<br />
Æ®Steinhaus© Ò‡‰appleÈÈËÒ<br />
* M. Martelli and G. Gannon, "Weighing Coins: Divide and Conquer to detect a counterfeit", College<br />
Mathematics Journal 28, 365-367, 1997.<br />
097<br />
‡ÈÏÈÏ‚<br />
ÈappleÂÈ<br />
≤∞∞∏
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