Шесть разноцветных гирь

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Сложность: 

Имеется 6 гирь: 2 красные, 2 белые и 2 синие. Внешне любая пара выглядит совершенно одинаково, но одна из гирь в каждой паре весит чуть больше другой. Все три более тяжёлые гири (красная, белая и синяя) имеют одинаковый вес. Все три более лёгкие гири также равны по весу.

Можно ли с помощью всего лишь двух взвешиваний на равноплечих весах определить, какая из гирь в каждой паре тяжелее?

Решение

Положим на одну чашу весов красную и белую гири, а на другую — синюю и белую гири.

Если чаши весов окажутся в равновесии, то вы будете знать, что на каждой чаше находится по одной тяжёлой и по одной лёгкой гире. Сняв с весов синюю и красную гири, вы оставите на чашах по белой гире и узнаете не только, какая из них тяжелее, но и какая из снятых гирь — красной и синей — была тяжёлой и какая лёгкой. Тем самым вы определите, какая из гирь, оставшихся в стороне, тяжёлая и какая лёгкая.

Если при первом взвешивании окажется, что одна чаша весов перетягивает другую, то вы сможете лишь утверждать, что белая гиря на опустившейся чаше тяжелее белой гири на поднявшейся чаше, но останетесь в неведении относительно веса красной и синей гирь.

Переложите обе белые гири на одну чашу весов, а красную и синюю гири — на другую (все четыре гири — те же, что и при первом взвешивании). Обозначим красную гирю буквой К, а синюю — буквой С. Возможны 3 исхода.

а) Чаша с двумя белыми гирями опустилась. Обе гири К и С на другой чаше весов лёгкие.

б) Чаша с гирями К и С опустилась. Обе гири К и С тяжёлые.

в) Чаши весов находятся в равновесии. Та из гирь К или С, которая при первом взвешивании находилась на опустившейся чаше, тяжёлая, а другая лёгкая.

Как и в первом случае, данные, полученные после двух взвешиваний, позволяют установить, какая из гирь, не подвергшихся взвешиванию, тяжёлая и какая лёгкая.




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