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1. The picture below shows a variant of a famous paradoxical puzzle. On the left, we take two rectangles of area 60, and cut each one into two pieces. On the right, we rearrange the four pieces, and put them together into a single rectangle of area 119. How could this be? 11 (a) Explain what's wrong. (Similar paradoxes can be found under names such as "Chessboard paradox" or "Missing square puzzle". It's fine to look at a source that explains such a paradox, provided you cite it and explain it in your own words.) (b) Although two 5 x 12 rectangles cannot really be rearranged into a 7 x 17 rectangle, it is possible to take two a × b rectangles and cut them as shown in the picture above to make a cxd rectangle, with no paradox. What should the lengths a, b, c, d be (up to scaling, of course)? (c) It is also possible to take three congruent rectangles, cut each one into two pieces, and rearrange them to form a single rectangle similar to the original three. How can we do this? Try to find an answer that lets you create a paradoxical decomposition of your own!

1. The picture below shows a variant of a famous paradoxical puzzle. On the left, we take two rectangles of area 60, and cut each one into two pieces. On the right, we rearrange the four pieces, and put them together into a single rectangle of area 119. How could this be? 11 (a) Explain what's wrong. (Similar paradoxes can be found under names such as "Chessboard paradox" or "Missing square puzzle". It's fine to look at a source that explains such a paradox, provided you cite it and explain it in your own words.) (b) Although two 5 x 12 rectangles cannot really be rearranged into a 7 x 17 rectangle, it is possible to take two a × b rectangles and cut them as shown in the picture above to make a cxd rectangle, with no paradox. What should the lengths a, b, c, d be (up to scaling, of course)? (c) It is also possible to take three congruent rectangles, cut each one into two pieces, and rearrange them to form a single rectangle similar to the original three. How can we do this? Try to find an answer that lets you create a paradoxical decomposition of your own!

Elementary Geometry For College Students, 7e
Elementary Geometry For College Students, 7e
7th Edition
ISBN:9781337614085
Author:Alexander, Daniel C.; Koeberlein, Geralyn M.
Publisher:Alexander, Daniel C.; Koeberlein, Geralyn M.
ChapterP: Preliminary Concepts
SectionP.CT: Test
Problem 1CT
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Please show what is wrong "mathematically". I don't mean you have to write a rigorous proof for everything, but please show what is going on with numbers and stuff

1. The picture below shows a variant of a famous paradoxical puzzle. On the left, we take two rectangles of area 60, and cut each one into two pieces. On the right, we rearrange the four pieces, and put them together into a single rectangle of area 119. How could this be? 11 (a) Explain what's wrong. (Similar paradoxes can be found under names such as "Chessboard paradox" or "Missing square puzzle". It's fine to look at a source that explains such a paradox, provided you cite it and explain it in your own words.) (b) Although two 5 x 12 rectangles cannot really be rearranged into a 7 x 17 rectangle, it is possible to take two a × b rectangles and cut them as shown in the picture above to make a cxd rectangle, with no paradox. What should the lengths a, b, c, d be (up to scaling, of course)? (c) It is also possible to take three congruent rectangles, cut each one into two pieces, and rearrange them to form a single rectangle similar to the original three. How can we do this? Try to find an answer that lets you create a paradoxical decomposition of your own!
Transcribed Image Text:1. The picture below shows a variant of a famous paradoxical puzzle. On the left, we take two rectangles of area 60, and cut each one into two pieces. On the right, we rearrange the four pieces, and put them together into a single rectangle of area 119. How could this be? 11 (a) Explain what's wrong. (Similar paradoxes can be found under names such as "Chessboard paradox" or "Missing square puzzle". It's fine to look at a source that explains such a paradox, provided you cite it and explain it in your own words.) (b) Although two 5 x 12 rectangles cannot really be rearranged into a 7 x 17 rectangle, it is possible to take two a × b rectangles and cut them as shown in the picture above to make a cxd rectangle, with no paradox. What should the lengths a, b, c, d be (up to scaling, of course)? (c) It is also possible to take three congruent rectangles, cut each one into two pieces, and rearrange them to form a single rectangle similar to the original three. How can we do this? Try to find an answer that lets you create a paradoxical decomposition of your own!
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