Number Play · Lesson 1 of 10
Numbers Tell us Things
“Discover how a short sequence can describe the order of people in a line.”
• Decode a height arrangement by counting taller people in front of each person. • Construct an arrangement that fits a given sequence of reports. • Use examples and counterexamples to judge statements about the arrangement. • Find the greatest report possible for a line of a given length.
A sequence can hide an arrangement
Imagine five children standing in a line, with the front of the line on the left. Each child says how many children standing before them are taller than they are. The reported numbers tell us something about their relative heights, even if we never measure a single child.
Only people in front count. A taller person behind a child does not affect that child’s number. The first child must say 0 because nobody is in front of them. As we move along the line, we compare each new child with every earlier child, not just with the person immediately ahead.
Problem
Five children have relative heights 3, 5, 2, 7, 4 from front to back. What do they say?
- 1.The first child says 0 because no one stands in front.
- 2.Height 5 is greater than 3, so the second child says 0. Height 2 is below both earlier heights, so the third says 2.
- 3.Height 7 is above all earlier heights, so the fourth says 0. Height 4 is below 5 and 7 but above 3 and 2, so the fifth says 2.
- 4.The sequence is 0, 0, 2, 0, 2.
Build a line from the numbers
We can also start with a sequence and try to make a height order. Work from left to right. At each place, the new person needs exactly the stated number of earlier people taller than them. Use distinct heights or cutouts and check every report after arranging them. Several arrangements may fit the same sequence.
Problem
Construct a line of four distinct heights that gives 0, 1, 2, 3.
- 1.The first report is necessarily 0.
- 2.The second person must be shorter than the first to say 1. The third must be shorter than both earlier people to say 2.
- 3.The fourth must be shorter than all three. Heights 4, 3, 2, 1 work.
Problem
Can heights 2, 4, 1, 3 produce the sequence 0, 0, 2, 1?
- 1.The first two say 0: nobody is ahead of the first, and height 4 is taller than height 2.
- 2.Height 1 has both 2 and 4 ahead and taller, so it says 2.
- 3.Only height 4 is ahead of and taller than height 3, so the last report is 1. The proposed sequence works.
Always, sometimes, or never?
A report of 0 means that no earlier person is taller. It does not mean the speaker is tallest in the whole line: someone taller could stand behind. The tallest person, however, must report 0, because nobody anywhere in the line is taller. This distinction shows why one example cannot prove an “always” claim, but one counterexample can disprove it.
The person with the largest report need not be the shortest. Their report counts taller people before them; a shorter person behind them could have a smaller report if fewer taller people stand ahead.
In a line of eight people, the last person has at most seven people in front. Thus no one can report more than 7. That value is possible if the last person is shorter than all seven earlier people. At position four, by the same reasoning, a report can be at most 3.
Quiz
The first person in any line must say which number?
A child says 0. Which conclusion is certain?
What sequence do distinct heights 4, 3, 2, 1 give from front to back?
What is the largest report possible among eight people?
If a child is tallest in the entire line, what must they report?
Practice Problems
- Find the sequence reported by distinct heights 5, 2, 4, 1, 3 from front to back. Explain each entry.
- Draw a line of five distinct heights whose reports are 0, 0, 0, 0, 0.
- Arrange five distinct heights to obtain 0, 1, 0, 1, 0. Check every position.
- Is the statement “anyone standing in the middle cannot say 0” always, sometimes, or never true? Give evidence.
- What is the greatest possible report at the sixth position in a line? Construct an example.
- Try a seven-person arrangement for 0, 1, 1, 2, 4, 1, 5. Explain how you checked it.
Key Takeaways
• Each report counts taller people in front of one person, not everyone taller in the group. • The first person and the tallest person each report 0, for different reasons. • A report at position n can be no greater than n − 1. • Build and check arrangements to test claims; use a counterexample to reject an “always” statement.
Previous
Start of chapter
Next · Lesson 2
Picking Parity