우리 곁에 있는 수학

Ⅰ 문제를 통한 수학적 사고의 이해
수학의 어원 ·············································································· 002
수학의 7가지 속성 ·································································· 003
① 핵심접근, 근본적 이해 ·············································· 003
쾨니히스베르크 다리 문제 ························································· 003
체스 문제 ·················································································· 005
기사의 여행 문제 ······································································ 006
오일러 그래프, 해밀턴 그래프 ···················································· 009
플뢰리 알고리즘 ········································································ 010
5개의 컵 문제 ··········································································· 011
물통 문제 ·················································································· 012
세 개의 산 문제 ········································································ 016
미로 찾기 ·················································································· 018
보드게임 SET ··········································································· 023
② 엄밀한 논리, 논증 ······························································ 030
플라톤 ······················································································ 031
윌리엄 블레이크의 〈태고의 나날들〉 ··········································· 032
폴 에이드리언 모리스 디랙 ························································ 033
겔만과 쿼크 그리고 군 ······························································· 034
자연과학의 속성 ········································································ 036
칼 포퍼 ····················································································· 038
유클리드와 원론 ········································································ 038
ㆍ 목 차 ㆍ
iv 목차
③ 창조성, 독창성, 다양성 ····················································· 043
피타고라스 정리의 증명(가필드의 증명법) ·································· 044
피타고라스 정리의 증명(박부성의 증명법) ·································· 045
피타고라스 정리의 증명(바스카라의 증명) ·································· 046
흥미로운 평면기하 문제들 ························································· 046
수학계의 흥미로운 논문들 ························································· 047
수학계의 흥미로운 미해결 문제들 ·············································· 056
④ 아름다움, 순수성 ································································ 059
수학의 아름다움에 대한 철학자, 수학자, 물리학자들의 주장 ······· 059
소수의 무한개수 증명 ································································ 062
테일러 전개식 ··········································································· 064
복소수의 아름다움 ···································································· 066
오일러 지표 ·············································································· 071
수학에서의 아름다움 ································································· 073
바젤 문제와 리만제타함수 ························································· 075
데자르그의 정리 ········································································ 079
파푸스 정리 ·············································································· 079
파스칼의 정리 ··········································································· 081
아름다운 수학공식 10선 ···························································· 081
⑤ 위태로운 벽돌 쌓기 ··························································· 083
러셀의 역설 ·············································································· 084
⑥ 무한대를 다룬다 ································································· 085
칸토어의 집합론 ········································································ 086
선택 공리 ·················································································· 089
연속체 가설 ·············································································· 090
괴델의 불완전성 정리 ································································ 091
목차 v
⑦ 유용성 ···················································································· 092
울릉도에서 독도가 보이는가? ···················································· 093
숫자 맞추기 ·············································································· 095
앨런 튜링 ·················································································· 101
루이 브라유 ·············································································· 101
모스부호 ··················································································· 103
만물은 수이다. ·········································································· 103
주민등록번호 ············································································ 105
성냥개비 게임 ··········································································· 106
아이스크림 31게임 ···································································· 107
국제수학올림피아드 ·································································· 108
게임이론 ··················································································· 110
초콜릿 게임 ·············································································· 111
신과의 치킨 게임 ······································································ 112
존 내쉬 ····················································································· 113
파푸스 정리 ·············································································· 113
몽주 ························································································· 114
수학적 모델링 ··········································································· 115
로트카-볼테라 모델 ·································································· 115
빅토리아 호수의 나일퍼치 이야기 ·············································· 117
빅토리아 호수의 부레옥잠 이야기 ·············································· 118
금융수학 ··················································································· 118
CT 사진의 원리 ········································································· 123
라돈 변환 ·················································································· 124
고속 푸리에 변환(FFT, Fast Fourier Transform) ························· 125
연습문제 ····················································································· 126
집합론 부록 ··············································································· 133
vi 목차
Ⅱ 생활속의 수학
수학과 영화 ·············································································· 144
마방진 ······················································································ 148
최석정과 라틴방진 ···································································· 154
오일러와 직교 라틴방진 ····························································· 157
지수귀문도 ················································································ 158
스도쿠 ······················································································ 159
마방진 난제 ·············································································· 162
샘 로이드 퍼즐 ·········································································· 165
미국의 3대 퍼즐 작가 ································································ 170
런던 지하철과 위상수학 ····························································· 171
회전 드럼 문제 ·········································································· 171
인간 vs 인공지능의 역사 ··························································· 171
행렬 ························································································· 178
Alpha max plus beta min algorithm ········································· 181
페이지 공식 ·············································································· 183
구글 입사 문제 ·········································································· 185
하노이의 탑 ·············································································· 192
체스판 위의 쌀 ·········································································· 194
어느 자선 사업가의 선행 ··························································· 194
케빈 베이컨의 6단계 ································································· 195
스탠리 밀그램의 실험 ································································ 196
A4 용지의 수학 ········································································· 196
빨간 풍선을 찾아라! ·································································· 202
복면산 ······················································································ 203
선교사와 식인종 문제 ································································ 208
목차 vii
필즈상(Fields Medal) ································································ 211
3대 수학상 ················································································ 213
브레이크스루상 ········································································· 214
경제학과 수학 ··········································································· 215
안정적인 결혼 문제 ··································································· 216
정보 이론 ·················································································· 220
논리 퍼즐 ·················································································· 221
부자의 유언 문제 ······································································ 229
간단한 숫자 마술 ······································································ 230
가중투표제 ················································································ 231
권력지수 ··················································································· 234
연습문제 ····················································································· 237
Ⅲ 생활속의 확률과 통계
확률이란 무엇인가 ···································································· 246
확률의 정의 ·············································································· 247
확률의 역사 ·············································································· 248
확률의 덧셈 법칙과 곱셈 법칙 ··················································· 252
확률에서 논란이 되는 문제 ························································ 253
오자의 개수 ·············································································· 257
보드게임 SET의 기댓값과 확률 ·················································· 257
확률을 이용한 정상과 비정상 구분하기 ······································ 260
1993년 LA 다저스 81승 81패 경기 결과 ······································ 261
동전 1000번 던지기 ·································································· 263
viii 목차
홀인원 확률 ·············································································· 264
남녀 폐암 발병 확률 ·································································· 266
로또 확률과 기댓값 ··································································· 266
은하계 로또 게임 ······································································ 268
자신에게는 엄격하고 남에게는 너그러운 확률 ···························· 269
이기적 유전자 ··········································································· 271
정액 베팅과 정률 베팅 ······························································· 274
베이즈 공식 ·············································································· 275
코로나의 확률 ··········································································· 277
코로나 백신의 위험성은? 278
연습문제 ····················································································· 281
참고문헌································································································ 283?

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