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果物の数え方
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Neria Nighthawk
October 18, 2019
Science
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果物の数え方
ネット上に上がっている観察力を問う問題に対して、数学的観点でどう解釈されるのか考察しました。
Neria Nighthawk
October 18, 2019
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Transcript
Ռͷ͑ํ ͶΓ͋ 2019 10 ݄ 18 ͶΓ͋ Ռͷ͑ํ 2019 10 ݄ 18 1 / 11
ࣗݾհ ໊લ ͶΓ͋ʢ@nerianighthawkʣ ུྺ ֶઐͷम࢜՝ఔଔۀޙɺIT اۀʹΤϯδχΞͱͯ͠ब৬ ઐ ཧཧֶɺݍ ͶΓ͋ Ռͷ͑ํ 2019 10 ݄ 18 2 / 11
ʁʹೖΔΛ͑ͳ͍͞ ͶΓ͋ Ռͷ͑ํ 2019 10 ݄ 18 3 / 11
ʁʹೖΔΛ͑ͳ͍͞ օ͞Μɺ͍ͭ͘ʹͳΔͱࢥ͍·͔͢ʁ࣍ͷϖʔδͰͪΐ ͬͱߟ͑ͯݟ·͠ΐ͏ɻ ͶΓ͋ Ռͷ͑ํ 2019 10 ݄ 18 3 / 11
ύλʔϯ 1 ·ͣɺΓΜ͕͝ 3 ͭͰ 30 ͳͷͰɺΓΜ͝ 1 ͭͰ 10 Ͱ͢Ͷɻ ΓΜ͝ 1 ͭɺόφφ 2 ͭͰ 18 ͳͷͰɺόφφ 1 ͭͰ 4ɻ ࣍ʹɺόφφ͔ΒίίφοπΛҾ͍ͯ 2 ͱͳ͍ͬͯΔͷͰɺίίφοπ͕ 2 ࠷ޙʹίίφοπɺΓΜ͝ɺόφφΛͯ͠ 16 ͶΓ͋ Ռͷ͑ํ 2019 10 ݄ 18 4 / 11
ύλʔϯ 1 ·ͣɺΓΜ͕͝ 3 ͭͰ 30 ͳͷͰɺΓΜ͝ 1 ͭͰ 10 Ͱ͢Ͷɻ ΓΜ͝ 1 ͭɺόφφ 2 ͭͰ 18 ͳͷͰɺόφφ 1 ͭͰ 4ɻ ࣍ʹɺόφφ͔ΒίίφοπΛҾ͍ͯ 2 ͱͳ͍ͬͯΔͷͰɺίίφοπ͕ 2 ࠷ޙʹίίφοπɺΓΜ͝ɺόφφΛͯ͠ 16 ͔͠͠ɺΑ͘ݟΔͱɺ࠷ޙͷίίφοπͱόφφࠓ·Ͱग़ͯདྷͨίίφοπόφφͱ ͕ҧ͍·͢Ͷʁ ͶΓ͋ Ռͷ͑ํ 2019 10 ݄ 18 4 / 11
ύλʔϯ 1 ·ͣɺΓΜ͕͝ 3 ͭͰ 30 ͳͷͰɺΓΜ͝ 1 ͭͰ 10 Ͱ͢Ͷɻ ΓΜ͝ 1 ͭɺόφφ 2 ͭͰ 18 ͳͷͰɺόφφ 1 ͭͰ 4ɻ ࣍ʹɺόφφ͔ΒίίφοπΛҾ͍ͯ 2 ͱͳ͍ͬͯΔͷͰɺίίφοπ͕ 2 ࠷ޙʹίίφοπɺΓΜ͝ɺόφφΛͯ͠ 16 ͔͠͠ɺΑ͘ݟΔͱɺ࠷ޙͷίίφοπͱόφφࠓ·Ͱग़ͯདྷͨίίφοπόφφͱ ͕ҧ͍·͢Ͷʁ ύλʔϯ 2 ͱ͍͏͜ͱͰɺ͖ͬ͞ͷύλʔϯ 1 ͱൺͯɺίίφοπɺόφφ 4 ຊ͕ 3 ຊʹ ͳ͍ͬͯΔͨΊͦΕͧΕ 1 2 , 3 4 ͯ݁͠Ռ 14 ͶΓ͋ Ռͷ͑ํ 2019 10 ݄ 18 4 / 11
ύλʔϯ 1 ·ͣɺΓΜ͕͝ 3 ͭͰ 30 ͳͷͰɺΓΜ͝ 1 ͭͰ 10 Ͱ͢Ͷɻ ΓΜ͝ 1 ͭɺόφφ 2 ͭͰ 18 ͳͷͰɺόφφ 1 ͭͰ 4ɻ ࣍ʹɺόφφ͔ΒίίφοπΛҾ͍ͯ 2 ͱͳ͍ͬͯΔͷͰɺίίφοπ͕ 2 ࠷ޙʹίίφοπɺΓΜ͝ɺόφφΛͯ͠ 16 ͔͠͠ɺΑ͘ݟΔͱɺ࠷ޙͷίίφοπͱόφφࠓ·Ͱग़ͯདྷͨίίφοπόφφͱ ͕ҧ͍·͢Ͷʁ ύλʔϯ 2 ͱ͍͏͜ͱͰɺ͖ͬ͞ͷύλʔϯ 1 ͱൺͯɺίίφοπɺόφφ 4 ຊ͕ 3 ຊʹ ͳ͍ͬͯΔͨΊͦΕͧΕ 1 2 , 3 4 ͯ݁͠Ռ 14 ͰɺͦͦՌͷຊΛ҆ʹֻ͚ࢉͰߟ͑ΔͷͲ͏ͳͷͰ͠ΐ͏͔ʁຊͷൺΛ ΓཱͨͤΔͷͰ͋ΕͳΜͰ͍͍ؾ͕͠·͢ɻ ͶΓ͋ Ռͷ͑ํ 2019 10 ݄ 18 4 / 11
ύλʔϯ 1 ·ͣɺΓΜ͕͝ 3 ͭͰ 30 ͳͷͰɺΓΜ͝ 1 ͭͰ 10 Ͱ͢Ͷɻ ΓΜ͝ 1 ͭɺόφφ 2 ͭͰ 18 ͳͷͰɺόφφ 1 ͭͰ 4ɻ ࣍ʹɺόφφ͔ΒίίφοπΛҾ͍ͯ 2 ͱͳ͍ͬͯΔͷͰɺίίφοπ͕ 2 ࠷ޙʹίίφοπɺΓΜ͝ɺόφφΛͯ͠ 16 ͔͠͠ɺΑ͘ݟΔͱɺ࠷ޙͷίίφοπͱόφφࠓ·Ͱग़ͯདྷͨίίφοπόφφͱ ͕ҧ͍·͢Ͷʁ ύλʔϯ 2 ͱ͍͏͜ͱͰɺ͖ͬ͞ͷύλʔϯ 1 ͱൺͯɺίίφοπɺόφφ 4 ຊ͕ 3 ຊʹ ͳ͍ͬͯΔͨΊͦΕͧΕ 1 2 , 3 4 ͯ݁͠Ռ 14 ͰɺͦͦՌͷຊΛ҆ʹֻ͚ࢉͰߟ͑ΔͷͲ͏ͳͷͰ͠ΐ͏͔ʁຊͷൺΛ ΓཱͨͤΔͷͰ͋ΕͳΜͰ͍͍ؾ͕͠·͢ɻ ύλʔϯ 3 ࠓՌͷΛࢦͩͱߟ͑ͯݟ·͠ΐ͏ɻ ͢Δͱόφφ 3 ຊ 2 √ 2ɺίίφοπ √ 2 ͱͳΔͷͰ݁Ռ 10 + 3 √ 2 ͱ͍͏͜ͱ ʹͳΓ·͢ɻ ͶΓ͋ Ռͷ͑ํ 2019 10 ݄ 18 4 / 11
ଞʹ͍͔ͭ͘ͷύλʔϯ͕ߟ͑ΒΕ·͢ɻͦͦ + ͷه߸ຊʹ͠ࢉͳͷ͔ͳͲ ͳͲɻ ͶΓ͋ Ռͷ͑ํ 2019 10 ݄ 18 5 / 11
ଞʹ͍͔ͭ͘ͷύλʔϯ͕ߟ͑ΒΕ·͢ɻͦͦ + ͷه߸ຊʹ͠ࢉͳͷ͔ͳͲ ͳͲɻ ͜͜Ͱɺৗࣝతʹֶ͔ͭతʹߟ͍͖͍͑ͯͨͱࢥ͍·͢ɻ ·ͣઌ΄Ͳͷύλʔϯ 3 φϯηϯεͰ͢ɻͳͥͳΒύλʔϯ 3 ͷ߹ɺόφφ 0 ຊ 0 Λҙຯ͢ΔͷͰ͕͑ 1 ʹͳΓ·͢ɻ ࣍ʹ + ͷه߸Γ͠ࢉͩͱߟ͑Δͷ͕ৗࣝతͰ͠ΐ͏ɻ ͶΓ͋ Ռͷ͑ํ 2019 10 ݄ 18 5 / 11
ଞʹ͍͔ͭ͘ͷύλʔϯ͕ߟ͑ΒΕ·͢ɻͦͦ + ͷه߸ຊʹ͠ࢉͳͷ͔ͳͲ ͳͲɻ ͜͜Ͱɺৗࣝతʹֶ͔ͭతʹߟ͍͖͍͑ͯͨͱࢥ͍·͢ɻ ·ͣઌ΄Ͳͷύλʔϯ 3 φϯηϯεͰ͢ɻͳͥͳΒύλʔϯ 3 ͷ߹ɺόφφ 0 ຊ 0 Λҙຯ͢ΔͷͰ͕͑ 1 ʹͳΓ·͢ɻ ࣍ʹ + ͷه߸Γ͠ࢉͩͱߟ͑Δͷ͕ৗࣝతͰ͠ΐ͏ɻ Ռͷଊ͑ํ 1 ͜͜·Ͱߟ͑Εɺ͋ͱՌΛͲ͏ଊ͑Δ͔ͱ͍͏͜ͱʹͳΓ·͢ɻ ͬͱ୯७ͳଊ͑ํՌΛมͱͯ͠ଊ͑Δ͜ͱͱͳΓ·͢ɻͦ͏ߟ͑ΔͱɺࣜҎ ԼͷΑ͏ͳ࿈ཱํఔࣜͩͱߟ͑Δ͜ͱ͕Ͱ͖·͢ɻ x + x + x = 30 x + y + y = 18 y − z = 2 z′ + x + y′ =? ͜ͷΑ͏ʹݟΔͱɺy ͱ y′ɺz ͱ z′ ʹؔੑ͕ͳ͍ͷͰɺ? ҙͷͱ͍͏͜ͱʹͳ Γ·͢ɻ ͶΓ͋ Ռͷ͑ํ 2019 10 ݄ 18 5 / 11
ଞʹ͍͔ͭ͘ͷύλʔϯ͕ߟ͑ΒΕ·͢ɻͦͦ + ͷه߸ຊʹ͠ࢉͳͷ͔ͳͲ ͳͲɻ ͜͜Ͱɺৗࣝతʹֶ͔ͭతʹߟ͍͖͍͑ͯͨͱࢥ͍·͢ɻ ·ͣઌ΄Ͳͷύλʔϯ 3 φϯηϯεͰ͢ɻͳͥͳΒύλʔϯ 3 ͷ߹ɺόφφ 0 ຊ 0 Λҙຯ͢ΔͷͰ͕͑ 1 ʹͳΓ·͢ɻ ࣍ʹ + ͷه߸Γ͠ࢉͩͱߟ͑Δͷ͕ৗࣝతͰ͠ΐ͏ɻ Ռͷଊ͑ํ 1 ͜͜·Ͱߟ͑Εɺ͋ͱՌΛͲ͏ଊ͑Δ͔ͱ͍͏͜ͱʹͳΓ·͢ɻ ͬͱ୯७ͳଊ͑ํՌΛมͱͯ͠ଊ͑Δ͜ͱͱͳΓ·͢ɻͦ͏ߟ͑ΔͱɺࣜҎ ԼͷΑ͏ͳ࿈ཱํఔࣜͩͱߟ͑Δ͜ͱ͕Ͱ͖·͢ɻ x + x + x = 30 x + y + y = 18 y − z = 2 z′ + x + y′ =? ͜ͷΑ͏ʹݟΔͱɺy ͱ y′ɺz ͱ z′ ʹؔੑ͕ͳ͍ͷͰɺ? ҙͷͱ͍͏͜ͱʹͳ Γ·͢ɻ ͡Ό͋ y′ = 3 4 y, z′ = 1 2 z ͱ͍͏ࣜΛ૿ͨ͠Β͍͍ͷͰʁͱࢥ͏͔͠Ε·ͤΜ͕ɺࣜ Λ૿͢ͳΜͯ͜ͱਆ༷͕Δ͜ͱͰɺΘΕΘΕҰൠਓ͕Δ͜ͱͰ͋Γ·ͤΜɻ ͶΓ͋ Ռͷ͑ํ 2019 10 ݄ 18 5 / 11
Ռͷଊ͑ํ 2 ࣍ՌΛఆͱͯ͠ଊ͑ͯΈ·͠ΐ͏ɻͭ·Γɺશମͷू߹ K ΛՌશମͷू߹ F ͱ ࣮શମͷू߹ R ͷू߹ͩͱߟ͑ͯɺͦ͜Ͱԋࢉ͕ߦΘΕ͍ͯΔͱ͠·͠ΐ͏ɻ a + a + a = 30 a + b + b = 18 b − c = 2 c′ + a + b′ =? ͜ͷ··ͩͱ͖ͬ͞ͱมΘΓ·ͤΜ͕Α͘ݟ͍ͯͩ͘͞ɻ ͶΓ͋ Ռͷ͑ํ 2019 10 ݄ 18 6 / 11
Ռͷଊ͑ํ 2 ࣍ՌΛఆͱͯ͠ଊ͑ͯΈ·͠ΐ͏ɻͭ·Γɺશମͷू߹ K ΛՌશମͷू߹ F ͱ ࣮શମͷू߹ R ͷू߹ͩͱߟ͑ͯɺͦ͜Ͱԋࢉ͕ߦΘΕ͍ͯΔͱ͠·͠ΐ͏ɻ a + a + a = 30 a + b + b = 18 b − c = 2 c′ + a + b′ =? ͜ͷ··ͩͱ͖ͬ͞ͱมΘΓ·ͤΜ͕Α͘ݟ͍ͯͩ͘͞ɻ ΓΜ͝ apple ͷ಄จࣈͰ a, όφφ banana Ͱ bɺίίφοπ coconut Ͱ c ʹͳͬͯ ͍·͢ʂ ͶΓ͋ Ռͷ͑ํ 2019 10 ݄ 18 6 / 11
Ռͷଊ͑ํ 2 ࣍ՌΛఆͱͯ͠ଊ͑ͯΈ·͠ΐ͏ɻͭ·Γɺશମͷू߹ K ΛՌશମͷू߹ F ͱ ࣮શମͷू߹ R ͷू߹ͩͱߟ͑ͯɺͦ͜Ͱԋࢉ͕ߦΘΕ͍ͯΔͱ͠·͠ΐ͏ɻ a + a + a = 30 a + b + b = 18 b − c = 2 c′ + a + b′ =? ͜ͷ··ͩͱ͖ͬ͞ͱมΘΓ·ͤΜ͕Α͘ݟ͍ͯͩ͘͞ɻ ΓΜ͝ apple ͷ಄จࣈͰ a, όφφ banana Ͱ bɺίίφοπ coconut Ͱ c ʹͳͬͯ ͍·͢ʂ ͱ͍͏ஊஔ͍͓͍ͯͯɺͰมͱఆͰԿ͕ҧ͏ͷͰ͠ΐ͏͔ʁ ม ্࣮ͷ࿈ཱํఔࣜʹݟ͑ΔΘ͚Ͱ͢ɻ͔͠͠ɺͦΕҎ্ͷ݅Λ͢Θ͚ʹ͍͔ ͣɺ͑ҙͷ࣮ΛऔΓ͏Δͱ͍͏͜ͱʹͳΓ·͢ɻ ఆ ू߹ K ԋࢉͰด͍ͯ͡Δͱߟ͑ΒΕΔͷͰ (ݫີʹຊʹԋࢉͰด͍ͯ͡Δ͔ ͷ͕ٞඞཁͰ͢)ɺ? ʹೖΔՌ͔࣮ͱ͍͏͜ͱʹͳΓ·͢ɻՌ͔࣮ͱ͍ ͏͜ͱͰɺมͷ߹ʹ͘Εͯൣғ͕͕͍ͬͯΔΑ͏ʹݟ͑·͕͢ɺͬͪ͜Ͱ উखʹ݅Λ͚Ճ͑Δ͜ͱ͕ՄೳͰ͢ɻ ͶΓ͋ Ռͷ͑ํ 2019 10 ݄ 18 6 / 11
ಉؔ ∼ ͕ಉؔͱҎԼͷੑ࣭Λຬͨ͢ͷͰ͢ɻ a ∼ a a ∼ b ⇒ b ∼ a a ∼ b ∧ b ∼ c ⇒ a ∼ c ಉྨͱू߹ શମू߹ S ʹಉؔ ∼ ͕ߟ͑ΒΕΔͱ͖ɺ͋Δݩ a ∈ S ͷಉྨ [a] ͱ [a] := {x ∈ S|x ∼ a} ·ͨɺू߹ S/ ∼ ͱ S := {[x]|x ∈ S} ͶΓ͋ Ռͷ͑ํ 2019 10 ݄ 18 7 / 11
ύλʔϯ 1 ͷ߹ Ռͷଊ͑ํΛఆͱͯ͠ଊ͑ͨ߹ɺύλʔϯ 1 ͷղ 16 ͱ͍͏ͷ͍͏ͷҎԼͷߟ͑ ํͰಋ͔ΕΔͣͰ͢ɻ ·ͣՌશମͷू߹ F ͷಉؔ ∼ Λಉ͡ՌͰ݁Δͷͱ͠·͢ɻྫ͑ 4 ຊͷό φφ b ͱ 3 ຊͷόφφ b′ ͲͬͪόφφͳͷͰ b ∼ b′ Ͱ͢ɻ ͦͯ͠ɺK′ := (F/ ∼) ∪ R ͱͯ͠ɺݩͷࣜΛߟ͑Δͱ [a] + [a] + [a] = 30 [a] + [b] + [b] = 18 [b] − [c] = 2 [c′] + [a] + [b′] =? ಉؔͷఆ͔ٛΒ [b] = [b′], [c] = [c′] Ͱ͢ɻ+ ͷه߸࣮ύʔτͰී௨ͷ͠ࢉΛ ҙຯ͍ͯ͠Δ͜ͱʹ͍ͯͨ͠ͷͰɺ[a] = 10, [b] = 4, [c] = 2 ͕Θ͔Γɺ? ͕ 16 Ͱ͋Δ͜ͱ ͕Θ͔Γ·͢ɻ ͶΓ͋ Ռͷ͑ํ 2019 10 ݄ 18 8 / 11
ύλʔϯ 2 ͷ߹ ՌͷຊΛֶʹམͱ͢ͷ͔ͳΓ͍͠ɻ ߟ͑ํͱͯ͠ɺͦΕͧΕͷՌʹ͕ଘࡏ͢Δͱߟ͑ͯɺͦͷΛຊͱ͢ΕΑ ͍ɻͭ·Γɺؔ f : F → R ͱͯ͠ɺՌ͕༩͑ΒΕͨͱ͖ͦͷՌͷݸɺຊΛ༩͑ Δؔͱ͢ΕΑ͍ɻ ྫ͑ b Λ 4 ຊͷόφφɺb′ Λ 3 ຊͷόφφͱ͢Δͱউखʹ 1 f(b) b = 1 f(b′) b′ ͱ͍͏͕ࣜ Γཱͭ͜ͱͱ͢ΕΑ͍ɻ ͢Δͱઌ΄Ͳͷࣜ a + a + a = 30 a + b + b = 18 b − c = 2 c′ + a + b′ =? ʹՃ͑ͯ 3b = 4b′, c = 2c′ ͱ͍͏ࣜߟ͑Δ͜ͱ͕Ͱ͖Δɻ݁Ռɺ? 14 ͱͳΔɻ ͶΓ͋ Ռͷ͑ํ 2019 10 ݄ 18 9 / 11
օ͞Μͷճ͍ͭ͘ʹͳΓ·͔ͨ͠ʁ ͶΓ͋ Ռͷ͑ํ 2019 10 ݄ 18 10 / 11
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օ͞Μͷճ͍ͭ͘ʹͳΓ·͔ͨ͠ʁ ͜Ε͕͑Կ͔ΛܾΊΔͨΊʹ͍ͬͯΔͷͰ͋Γ·ͤΜɻ Ή͠Ζ͜ͷʹରͯ͠ʮ16 ؒҧ͍Ͱਖ਼ղ 14 Ͱ͢ʯΈ͍ͨʹܾΊ͚ͭͯ͠·͏͜ͱ ͷํ͕ؒҧ͍ͩͱࢥ͍·͢ɻ ͔͠͠ɺ͕͑ 16 ͩͱࢥ͏͔Βʹͦ͏ࢥ͏ཧ༝͕͋ΔͣͰ͢ɻֶΛษڧ͍ͯ͠Ε ɺͦͷཧ༝Λଞͷਓʹ͏·͘આ໌Ͱ͖Δͱࢥ͍·͢ɻ ͶΓ͋ Ռͷ͑ํ 2019 10 ݄ 18 10 / 11
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