Problem 1 · 2022 Math Kangaroo
Medium
Spatial & Visual Reasoning
sequence-of-figures
Martin's smartphone displays the diagram on the right. It shows how long he has worked with four different apps in the previous week. The apps are sorted from top to bottom according to the amount of time they have been used. This week he has spent only half the amount of time using two of the apps and the same amount of time as last week using the other two apps. Which of the following pictures cannot be the diagram for the current week?
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Answer: E
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Hint 1 of 3
Each app's new bar is either the same length as last week's or exactly half of it.
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Hint 2 of 3
Two bars must stay unchanged and the other two must be halved — look for the picture that can't be built that way.
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Hint 3 of 3
The total of all four new bars is fixed; the picture whose bars can't be split into two 'kept' and two 'halved' originals is the answer.
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Approach: rule out the bar chart that can't come from keeping two bars and halving two
- Last week's four bars had fixed (decreasing) lengths; this week exactly two of them keep their length and the other two are cut to half.
- So every valid new picture must show two bars equal to two of the originals and two bars equal to half of the other two originals.
- Checking each option, four can be matched to such a 'keep two, halve two' pairing, but one cannot — it has a bar that is neither a full original nor half of any original.
- That impossible diagram is E.
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