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(g) (cm)

(g) (cm) 5 6.25 7.5 8.75 10 90 112.5 135 157.5 180 ● ● ● ● ● ● ● ● ● ● (g) (cm) む む

(g) (cm) 5 6.25 7.5 8.75 10 90 112.5 135 157.5 180 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● (g) (cm) む む

(g) (cm) 5 6.25 7.5 8.75 10 90 112.5 135 157.5 180 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● (g) (cm) む む

(g) (cm) 5 6.25 7.5 8.75 10 90 112.5 135 157.5 180 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● (g) (cm) む む

(g) (cm) 5 6.25 7.5 8.75 10 90 112.5 135 157.5 180 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● (g) (cm) む む

(g) (cm) 5 6.25 7.5 8.75 10 90 112.5 135 157.5 180 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● (g) (cm) む む

(g) (cm) 5 6.25 7.5 8.75 10 90 112.5 135 157.5 180 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● (g) (cm) む む

(g) (cm) 5 6.25 7.5 8.75 10 90 112.5 135 157.5 180 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 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● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 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● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● (g) (cm) む む

(g) (cm) 5 6.25 7.5 8.75 10 90 112.5 135 157.5 180 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 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● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● (g) (cm) む む

(g) (cm) 5 6.25 7.5 8.75 10 90 112.5 135 157.5 180 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 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● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● (g) (cm) む む (cm) (g) む む

む む or 5 6.25 7.5 8.75 10 90 112.5 135 157.5 180 ● ● ● ● ● ● ● ● ● ● (g) (cm) (cm) (g) む む

む む or 5 6.25 7.5 8.75 10 90 112.5 135 157.5 180 ● ● ● ● ● ● ● ● ● ● (g) (cm) (cm) (g) む む

む む or 5 6.25 7.5 8.75 10 90 112.5 135 157.5 180 ● ● ● ● ● ● ● ● ● ● (g) (cm) (cm) (g) む む

む む or 5 6.25 7.5 8.75 10 90 112.5 135 157.5 180 ● ● ● ● ● ● ● ● ● ● (g) (cm) (cm) (g) む む

む む or 5 6.25 7.5 8.75 10 90 112.5 135 157.5 180 ● ● ● ● ● ● ● ● ● ● (g) (cm) (cm) (g) む む

む む or 5 6.25 7.5 8.75 10 90 112.5 135 157.5 180 ● ● ● ● ● ● ● ● ● ● (g) (cm) (cm) (g) む む

む む or 5 6.25 7.5 8.75 10 90 112.5 135 157.5 180 ● ● ● ● ● ● ● ● ● ● (g) (cm) (cm) (g) む む

む む or 5 6.25 7.5 8.75 10 90 112.5 135 157.5 180 ● ● ● ● ● ● ● ● ● ● (g) (cm) (cm) (g) む む

( ) J aime la musique I love music

Deep Fake "MQIB(P "MQIB'PME "MQIB5FOTPS ChatGPT

GPT n 稿 (Lossy )

Decision Tree Random Forest GBDT Nearest Neighbor Logistic Regression SVM Gaussian Process Neural Network

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0 1 P( =red)

Random Forest Gaussian Process Logistic Regression 0 1 P( =red)

Random Forest Neural Network SVR Kernel Ridge p1 p2 p3 p4

(Bioinformatics) Materials Informatics ( ) 稿

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ダ ダ (data leakage) テ Training ( ) デ Validation ( ) =

ダ ダ (data leakage) テ Training ( ) デ Validation ( ) =

ダ ダ (data leakage) テ Training ( ) デ Validation ( ) ト Test ( ) =

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 y ‣ (= ) (x , x ) y ( )

( ) 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 x1 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Fisher ( ) . (Replication) ⾒ ( ) . (Local Control) . (Randomization) (Fisher ) Data Leakage

:

: ( ) ⾒ ༗઒અ෉ Herbert Simon Simon HA, Machine Discovery. ( ) Langley PW, Simon HA, Bradshaw G, Zytkow JM, Scienti c Discovery: Computational Explorations of the Creative Process ( ). , . (1996) , . (2001)

: ( ) ⾒ ༗઒અ෉ Herbert Simon Simon HA, Machine Discovery. ( ) Langley PW, Simon HA, Bradshaw G, Zytkow JM, Scienti c Discovery: Computational Explorations of the Creative Process ( ). , . (1996) , . (2001)

3 . . ( ) .

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 x AAAB93icbVDLSgNBEOyNrxhfUY9eFoPgKeyKr2PQi8cEzAOSJcxOepMhM7PLzKywhHyBVz17E69+jkf/xEmyBxMtaCiquunuChPOtPG8L6ewtr6xuVXcLu3s7u0flA+PWjpOFcUmjXmsOiHRyJnEpmGGYydRSETIsR2O72d++wmVZrF8NFmCgSBDySJGibFSI+uXK17Vm8P9S/ycVCBHvV/+7g1imgqUhnKiddf3EhNMiDKMcpyWeqnGhNAxGWLXUkkE6mAyP3Tqnlll4EaxsiWNO1d/T0yI0DoToe0UxIz0qjcT//O6qYlugwmTSWpQ0sWiKOWuid3Z1+6AKaSGZ5YQqpi91aUjogg1NpulLaGY2kz81QT+ktZF1b+uXjUuK7W7PJ0inMApnIMPN1CDB6hDEyggPMMLvDqZ8+a8Ox+L1oKTzxzDEpzPH5Ack50= y A.

Two Cultures: Leo Breiman, Statistical Modeling: The Two Cultures. (with comments and a rejoinder by the author), Statistical Science, 16(3): 199-231 (August 2001) https://doi.org/10.1214/ss/1009213726

Two Cultures: Data Modeling Algorithmic Modeling 2 Leo Breiman, Statistical Modeling: The Two Cultures. (with comments and a rejoinder by the author), Statistical Science, 16(3): 199-231 (August 2001) https://doi.org/10.1214/ss/1009213726

Two Cultures: Data Modeling Algorithmic Modeling 2 ( ) Leo Breiman, Statistical Modeling: The Two Cultures. (with comments and a rejoinder by the author), Statistical Science, 16(3): 199-231 (August 2001) https://doi.org/10.1214/ss/1009213726

Two Cultures: Data Modeling Algorithmic Modeling 2 ( ) CP Leo Breiman, Statistical Modeling: The Two Cultures. (with comments and a rejoinder by the author), Statistical Science, 16(3): 199-231 (August 2001) https://doi.org/10.1214/ss/1009213726

Two Cultures: Data Modeling Algorithmic Modeling 2 ( ) CP 2001 20 Leo Breiman, Statistical Modeling: The Two Cultures. (with comments and a rejoinder by the author), Statistical Science, 16(3): 199-231 (August 2001) https://doi.org/10.1214/ss/1009213726

Two Cultures: Data Modeling Algorithmic Modeling 2 ( ) CP 2001 20 Cox Efron ( ) Leo Breiman, Statistical Modeling: The Two Cultures. (with comments and a rejoinder by the author), Statistical Science, 16(3): 199-231 (August 2001) https://doi.org/10.1214/ss/1009213726

Two Cultures: Data Modeling Algorithmic Modeling 2 ( ) CP 2001 20 Cox Efron ( ) 20話 2021 Observational Studies Leo Breiman, Statistical Modeling: The Two Cultures. (with comments and a rejoinder by the author), Statistical Science, 16(3): 199-231 (August 2001) https://doi.org/10.1214/ss/1009213726

Two Cultures: Breiman's main point is: If you want prediction, do prediction for its own sake and forget about the illusion of representing nature. (Judea Pearl) https://doi.org/10.1353/obs.2021.0008

Two Cultures: Breiman's main point is: If you want prediction, do prediction for its own sake and forget about the illusion of representing nature. (Judea Pearl) https://doi.org/10.1353/obs.2021.0008 Three Lessons There have been particularly exciting developments in the last five years. 
 What has been learned? The three lessons that seem most important to one: Rashomon: the multiplicity of good models; Occam: the conflict between simplicity and accuracy; Bellman: dimensionality—curse or blessing.

Rashomon: the multiplicity of good models ΍Ϳʲᤵʳͷதʢͳ͔ʣ ᶄ (և઒ཾ೭հͷ࡞඼ʮᤵͷதʯ͔Β) ؔ܎ऀͷ͍͏͜ͱ͕৯͍ҧͬͯɺਅ૬͕Θ͔Βͳ͍ ͜ͱΛ͍͏ɻʮਅ૬͸ᤵͷதͩʯ(ਫ਼બ൛ ೔ຊࠃޠେࣙయΑΓ)ʢ Rashomon 1950

Rashomon: the multiplicity of good models ΍Ϳʲᤵʳͷதʢͳ͔ʣ ᶄ (և઒ཾ೭հͷ࡞඼ʮᤵͷதʯ͔Β) ؔ܎ऀͷ͍͏͜ͱ͕৯͍ҧͬͯɺਅ૬͕Θ͔Βͳ͍ ͜ͱΛ͍͏ɻʮਅ૬͸ᤵͷதͩʯ(ਫ਼બ൛ ೔ຊࠃޠେࣙయΑΓ)ʢ Rashomon 1950 Rashomon E ect ( ):

Rashomon: the multiplicity of good models ΍Ϳʲᤵʳͷதʢͳ͔ʣ ᶄ (և઒ཾ೭հͷ࡞඼ʮᤵͷதʯ͔Β) ؔ܎ऀͷ͍͏͜ͱ͕৯͍ҧͬͯɺਅ૬͕Θ͔Βͳ͍ ͜ͱΛ͍͏ɻʮਅ૬͸ᤵͷதͩʯ(ਫ਼બ൛ ೔ຊࠃޠେࣙయΑΓ)ʢ Rashomon 1950 Rashomon E ect ( ): ⾒ ( )

Rashomon: the multiplicity of good models ΍Ϳʲᤵʳͷதʢͳ͔ʣ ᶄ (և઒ཾ೭հͷ࡞඼ʮᤵͷதʯ͔Β) ؔ܎ऀͷ͍͏͜ͱ͕৯͍ҧͬͯɺਅ૬͕Θ͔Βͳ͍ ͜ͱΛ͍͏ɻʮਅ૬͸ᤵͷதͩʯ(ਫ਼બ൛ ೔ຊࠃޠେࣙయΑΓ)ʢ Rashomon 1950 Rashomon E ect ( ): ⾒ ( ) ( )

Rashomon: the multiplicity of good models Underspeci cation (specify ) ( )

Rashomon: the multiplicity of good models Underspeci cation (specify ) ( ) Underspeci cation Google

Occam: the conflict between simplicity and accuracy ResNet : 2600 ResNet : 4500 E cientNet-B : 6600 VGG : 1 4400 ⾒ 20×20 1200 1000×1000 300 -layer, -heads BERT: 1 1000 -layer, -heads BERT: 3 3600 GPT- XL: 15 5800 GPT- : 1750 ダ ( ) ダ

Occam: the conflict between simplicity and accuracy ResNet : 2600 ResNet : 4500 E cientNet-B : 6600 VGG : 1 4400 ⾒ 20×20 1200 1000×1000 300 -layer, -heads BERT: 1 1000 -layer, -heads BERT: 3 3600 GPT- XL: 15 5800 GPT- : 1750 ダ ( ) ダ ( )

Bellman: dimensionality curse or blessing ( ⾒ )

Bellman: dimensionality curse or blessing ( ⾒ ) 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 1 AAAChHichVHLSsNAFD2Nr/ps1Y3gRiwVF1JutD5wIUU3LltrVdAiSZzW0DQJSVqoxR/QreLClYIL8QP8ADf+gAs/QVxWcOPC2zQgWtQbJnPmzD13zsxVbUN3PaLnkNTR2dXdE+7t6x8YHIpEh0e2XKviaCKnWYbl7KiKKwzdFDlP9wyxYztCKauG2FZLa8397apwXN0yN72aLfJlpWjqBV1TPKYy8n40RgnyY6IdyAGIIYi0Fb3HHg5gQUMFZQiY8BgbUODytwsZBJu5POrMOYx0f1/gGH2srXCW4AyF2RL/i7zaDViT182arq/W+BSDh8PKCcTpiW6pQY90Ry/08Wutul+j6aXGs9rSCns/cjKWff9XVebZw+GX6k/PHgpY8r3q7N32meYttJa+enTRyC5vxOtTdE2v7P+KnumBb2BW37SbjNi4/MOPyl74xbhB8s92tIOt2YS8kEhmkrHUatCqMMYxiWnuxyJSWEcaOa4vcIoznEvd0ow0J823UqVQoBnFt5BWPgE/Y4+x 1 ) 1 1 ( d )

Bellman: dimensionality curse or blessing ( ⾒ ) 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 0.5 p d 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 1 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 1 ) 1 1 ( d )

Bellman: dimensionality curse or blessing ( ⾒ ) 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 0.5 p d 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 1 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 1 ) 1 1 ( d ) 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 d = 2 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 0.5 p 2

Bellman: dimensionality curse or blessing ( ⾒ ) 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 0.5 p d 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 1 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 1 ) 1 1 ( d ) 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 d = 2 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 0.5 p 2 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 d = 4 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 1

Bellman: dimensionality curse or blessing ( ⾒ ) AAACjnichVHNSgJRFD5Of2Y/Wm2CNpIYreQYZhFEUhuX/uQPqMjMeLPBcWaauQomvkD7aBEUBS2iB+gB2vQCLXyEaGnQpkXHcSBKsjPcud/97vnO/e49kqEqFkfsuoSx8YnJKfe0Z2Z2bt7rW1jMWnrDlFlG1lXdzEuixVRFYxmucJXlDZOJdUllOal20N/PNZlpKbp2yFsGK9XFqqYcKbLIiSpgaLNonZi8XemUfQEMoR3+YRB2QACcSOi+RyhCBXSQoQF1YKABJ6yCCBZ9BQgDgkFcCdrEmYQUe59BBzykbVAWowyR2Br9q7QqOKxG635Ny1bLdIpKwySlH4L4gvfYw2d8wFf8/LNW267R99KiWRpomVH2ni2nP/5V1WnmcPytGumZwxFs214V8m7YTP8W8kDfPL3opXdSwfYa3uIb+b/BLj7RDbTmu3yXZKnLEX4k8kIvRg0K/27HMMhuhMLRUCQZCcT2nVa5YQVWYZ36sQUxiEMCMvaLnsMVXAs+ISrsCnuDVMHlaJbgRwjxLwzik/k= 0.5 p d 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 1 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 1 ) 1 1 ( d ) 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 d = 2 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 0.5 p 2 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 d = 4 AAAChHichVHLSsNAFD2Nr/ps1Y3gRiwVF1JutD5wIUU3LltrVdAiSZzW0DQJSVqoxR/QreLClYIL8QP8ADf+gAs/QVxWcOPC2zQgWtQbJnPmzD13zsxVbUN3PaLnkNTR2dXdE+7t6x8YHIpEh0e2XKviaCKnWYbl7KiKKwzdFDlP9wyxYztCKauG2FZLa8397apwXN0yN72aLfJlpWjqBV1TPKYy8n40RgnyY6IdyAGIIYi0Fb3HHg5gQUMFZQiY8BgbUODytwsZBJu5POrMOYx0f1/gGH2srXCW4AyF2RL/i7zaDViT182arq/W+BSDh8PKCcTpiW6pQY90Ry/08Wutul+j6aXGs9rSCns/cjKWff9XVebZw+GX6k/PHgpY8r3q7N32meYttJa+enTRyC5vxOtTdE2v7P+KnumBb2BW37SbjNi4/MOPyl74xbhB8s92tIOt2YS8kEhmkrHUatCqMMYxiWnuxyJSWEcaOa4vcIoznEvd0ow0J823UqVQoBnFt5BWPgE/Y4+x 1 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 d > 4 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 0.5 p d

Bellman: dimensionality curse or blessing ( ⾒ ) 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 0.5 p d 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 1 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 1 ) 1 1 ( d ) 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 d = 2 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 0.5 p 2 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 d = 4 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 1 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 d > 4 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 0.5 p d 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 > 1 !?

Bellman: dimensionality curse or blessing ( ⾒ ) 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 0.5 p d 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 1 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 1 ) 1 1 ( d ) 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 d = 2 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 0.5 p 2 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 d = 4 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 1 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 d > 4 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 0.5 p d 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 > 1 !? 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 V (d) = ⇡ d 2 d 2 · d 2 ! 0 (d ! 1) = = d AAAB/XicbVDLSgNBEOz1GeMr6tHLYBA8hV3xdRGCXjxGMA9I1jA7O5sMmZldZmaFsAS/wauevYlXv8Wjf+Ik2YNJLGgoqrrp7goSzrRx3W9naXlldW29sFHc3Nre2S3t7Td0nCpC6yTmsWoFWFPOJK0bZjhtJYpiEXDaDAa3Y7/5RJVmsXwww4T6AvckixjBxkpN7zFE18jrlspuxZ0ALRIvJ2XIUeuWfjphTFJBpSEca9323MT4GVaGEU5HxU6qaYLJAPdo21KJBdV+Njl3hI6tEqIoVrakQRP170SGhdZDEdhOgU1fz3tj8T+vnZroys+YTFJDJZkuilKOTIzGv6OQKUoMH1qCiWL2VkT6WGFibEIzWwIxspl48wksksZpxbuonN+flas3eToFOIQjOAEPLqEKd1CDOhAYwAu8wpvz7Lw7H87ntHXJyWcOYAbO1y9Fl5UB 1d = 1

Bellman: dimensionality curse or blessing ⾒ ( )

Bellman: dimensionality curse or blessing ⾒ ( ) . (Lipschitz ) ⾒ ( ) d d= 100 (10 ) d= 10 =100

Bellman: dimensionality curse or blessing ⾒ ( ) . (Lipschitz ) ⾒ ( ) d d= 100 (10 ) d= 10 =100 ( ) . (d> )

3 . . ( ) .

Informatics ( ) ⾒

Informatics ( ) ⾒ ⾒ (Herbert Simon!) ⾒ ( )

Informatics ( ) ⾒ ⾒ (Herbert Simon!) ⾒ ( ) ⾒

( )

( ) X Y Y

( ) X Y Y

( ) X Y Y

vs

vs (= )

vs (= ) • “Theory-driven models can be wrong. But data-driven models cannot be wrong or right. Data-driven are not trying to describe an underlying reality.” David Hand, KDD2018 (Keynote Talk) http://videolectures.net/kdd2018_hand_data_science/

vs (= ) • “Theory-driven models can be wrong. But data-driven models cannot be wrong or right. Data-driven are not trying to describe an underlying reality.” David Hand, KDD2018 (Keynote Talk) http://videolectures.net/kdd2018_hand_data_science/ ⾒ ( )

⾒ Universal Approximation Theorem ⾒ ( ) Blackbox vs

⾒ ⾒ Universal Approximation Theorem ⾒ ( ) Blackbox vs

⾒ ⾒ Universal Approximation Theorem ⾒ ( ) Blackbox vs

Physics-informed ML, Geometric ML, Causal ML https://doi.org/10.1038/s42254-021-00314-5 × × ×

⾒ (NP-hard ) Schrittweiser et al. Mastering Atari, Go, chess and shogi by planning with a learned model. (“MuZero”, Nature, 2020) Davies et al. Advancing mathematics by guiding human intuition with AI. (Nature, 2021) Fawzi et al. Discovering faster matrix multiplication algorithms with reinforcement learning. (“AlphaTensor”, Nature, 2022) × closed /

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