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Lesson 1 of 8

Number Play · Lesson 1 of 8

Numbers as Clues and Supercells

“Use neighbouring values to decode arrangements and build number puzzles.”

Learning Objectives

• Explain how a number can describe information about an arrangement. • Count taller neighbours and justify possible or impossible height arrangements. • Identify supercells in a row and in a grid using the correct neighbours. • Construct rows with specified or maximum numbers of supercells. • Use largest and smallest values to explain why a construction works.

Numbers can describe more than a count

A number on a bus may identify a route, while a number on a measuring tape describes a length. The same symbol can carry different information depending on the rule being used. Before doing a calculation, ask what each number means. In a puzzle, discovering that meaning can be the main piece of work.

Think of five situations in which you use numbers: finding a house, reading a clock, paying for something, measuring your height, or counting people in a room. Explain what the number tells you in each situation. A house number of 45 does not mean that 45 people live there. Understanding the context prevents a perfectly correct calculation from answering the wrong question.

A line of children gives clues about height

Imagine children standing in a line. Each child says how many of the children immediately beside them are taller. A child in the middle compares with two neighbours; a child at either end has only one neighbour. Nobody compares with every child in the line. A very tall child several places away has no effect on the number you say.

Definition
Neighbour in a line

The person or cell immediately to the left or right. An end position has only one neighbour.

The possible numbers are 0, 1, and 2. Saying 0 means that no immediate neighbour is taller. Saying 1 means that exactly one is taller. Saying 2 means that both are taller. Someone of equal height is not taller, so equal heights must be treated carefully. For the following height constructions, use five children of different heights.

One arrangement of five different heights01210
Heights determine the neighbour counts— Read only the bars immediately beside each bar. Taller bars represent taller children; the figures beneath are the counts they say.
Example — Decoding one arrangement

Problem
Five children have heights ordered as 4, 3, 1, 2, 5, where the numbers represent relative height. What does each say?

  1. 1.The first child has only height 3 beside them. Since 3 is not taller than 4, the first count is 0.
  2. 2.The second child has heights 4 and 1 beside them. Only 4 is taller than 3, so the count is 1.
  3. 3.The middle child has heights 3 and 2 beside them. Both are taller than 1, so the count is 2.
  4. 4.The fourth child has heights 1 and 5 beside them. Only 5 is taller than 2, so the count is 1. The last child has no taller neighbour, so its count is 0.
  5. 5.The complete sequence is 0, 1, 2, 1, 0. Checking every position explains why it is possible.

An end child cannot say 2 because there is only one neighbour to count. A tallest child always says 0, wherever they stand. Consequently five children of different heights cannot all say 1. They also cannot all say 0: take any adjacent pair with different heights, and the shorter member has at least one taller neighbour. Equal-height children could all say 0, but rearranging children does not change their heights.

Example — Making as many counts of 2 as possible

Problem
How can five children of different heights stand so that as many as possible say 2?

  1. 1.Neither end can say 2. Among the three middle positions, adjacent children cannot both have two taller neighbours: each would have to be shorter than the other.
  2. 2.Use the relative height order 5, 1, 4, 2, 3. The second child is shorter than 5 and 4; the fourth is shorter than 4 and 3.
  3. 3.The spoken counts are 0, 2, 0, 2, 0. Two children say 2, and the neighbour restriction prevents three from doing so.
Investigate other arrangements

An increasing order of five different heights gives 1, 1, 1, 1, 0. Adjacent children can therefore say the same number. Try a decreasing order, and then rearrange the children to produce 0, 1, 2, 1, 0. Give the height order as evidence rather than simply answering yes.

Supercells are larger than their neighbours

Now replace the children with a row of numbered cells. Instead of counting taller neighbours, colour a cell when its number is greater than every number immediately beside it. We call such a cell a supercell. An end cell still has only one neighbour, so it can be a supercell even though it is not between two smaller values.

Definition
Supercell

A cell whose number is strictly greater than the numbers in all its neighbouring cells.

For a short initial check, consider 200, 577, 626, 345, 790, 694, 109, 198. The entry 626 exceeds 577 and 345, while 200 fails because its only neighbour 577 is larger. The last entry 198 qualifies by exceeding its only neighbour 109. The complete set of supercells is 626, 790, and 198. This comparison explains both interior and end cases.

A comparison with immediate neighbours682867094353780370873088000558352Greater means strictly greater, not equal
Find the peaks in a row— Green cells are supercells. The left end is checked against one neighbour; the other green cells exceed both neighbours.
Example — Checking a numbered row

Problem
Find the supercells in 6828, 670, 9435, 3780, 3708, 7308, 8000, 5583, 52.

  1. 1.At the left end, 6828 exceeds its only neighbour 670, so it is a supercell.
  2. 2.9435 exceeds both 670 and 3780. Likewise, 8000 exceeds 7308 and 5583. These are supercells.
  3. 3.7308 is larger than 3708 but smaller than 8000, so it fails the rule. Check the remaining entries in the same way.
  4. 4.The supercells are 6828, 9435, and 8000. A number can be large without being a supercell if an even larger number is beside it.

Two adjacent cells cannot both be supercells: the first would need to exceed the second, and the second would need to exceed the first. This gives a useful way to construct a row. Alternate a high value with a low value, starting and, if possible, ending with high values. To satisfy a specified colour pattern, first put large values into the intended supercells and then check every unwanted cell as well.

Example — Maximum supercells in nine positions

Problem
Fill nine cells using different numbers between 100 and 1000 to obtain as many supercells as possible.

  1. 1.Use 110, 100, 150, 130, 280, 200, 230, 210, 270. Positions 1, 3, 5, 7, and 9 exceed their neighbours.
  2. 2.There are five supercells. Six would require at least one pair of adjacent supercells, which is impossible.
  3. 3.For an even number of positions, pairing positions shows that at most half can be supercells. For an odd number, alternating allows one extra high position. Thus 2, 4, 6, 8 positions allow 1, 2, 3, 4; and 3, 5, 7, 9 allow 2, 3, 4, 5.

With distinct numbers in a row containing at least two cells, the largest number must be a supercell: every neighbour is smaller. The smallest cannot be one. A row without any supercell is therefore impossible under those conditions. The second largest can fail if it is beside the largest. Even the second smallest can be a supercell if it sits at an end beside the smallest; 2, 1, 3, 4, 5, 6, 7, 9, 8 demonstrates both observations.

Check the conditions before making a general claim

If equal numbers are allowed, the largest value need not be strictly greater than its neighbours. A row such as 8, 8, 8 has no supercell. The conclusions about an unavoidable largest supercell above rely on all entries being distinct.

In a grid, neighbours lie in four directions

A grid adds another direction of comparison. Check the cells directly left, right, above, and below; diagonal cells do not count. A corner has two neighbours, a non-corner edge has three, and an interior cell has four. The meaning of greater than every neighbour stays the same.

2430750073509870311547959124923045808632828034465785194458056034Check all four directions where neighbours exist
Supercells in a grid— The green entries exceed every existing neighbour immediately above, below, left, and right. Diagonals are excluded.

For example, 8632 exceeds 4580, 8280, 4795, and 1944. These four checks establish that it is a supercell. A large diagonal value would not change that conclusion. When filling a grid, it is easy to create an unwanted supercell, so use the neighbour rule to inspect the entire finished grid.

Column 1Column 2Column 3Column 4
96,31096,30136,10939,160
96,10313,60960,31919,306
13,90610,39660,19360,931
10,36910,96310,93669,031
A five-digit construction to investigate

Every entry uses 1, 0, 6, 3, and 9 once. Its supercells are at row 1 column 1, row 1 column 4, row 2 column 3, row 4 column 2, and row 4 column 4. Verify each and check all other cells. The largest entry is 96,310; the smallest even entry is 10,396; and the smallest entry above 50,000 is 60,193. Commas separate the thousands group from the final three digits. The zero can appear inside a five-digit number, but it cannot be its first digit.

Example — Choosing values for prescribed peaks

Problem
How can you choose some of the missing entries when 96,301, 36,109, 13,609, 60,319, 19,306, 60,193, and 10,963 are already fixed?

  1. 1.A required top-left peak must exceed its neighbour 96,301. The arrangement 96,310 does so, and its lower neighbour can be 96,103.
  2. 2.A required top-right peak must exceed 36,109 to its left and 19,306 below. The five-digit arrangement 39,160 works.
  3. 3.To keep the fixed 10,963 as a bottom-row peak, choose its other neighbours below it: 10,369, 10,936, and 10,396.
  4. 4.Complete the remaining entries and check every cell, including unwanted peaks. The table above gives one complete solution; there can be others.
Design a supercell challenge

Ask a partner to fill nine cells with exactly four supercells, or to show why six supercells are impossible. Change the number of positions or prescribe different peak positions. Solve your own version first and state whether repeated values are allowed so that the conditions are clear.

Quiz

Quick check

How many immediate neighbours does a child at an end of a line have?

Quick check

Five children have different heights. Why can they not all say 1 under the taller-neighbour rule?

Quick check

Which entries are supercells in the row 8, 3, 6, 9, 2?

Quick check

What is the maximum number of supercells in a row of seven distinct entries?

Quick check

Which cell is excluded when checking an interior supercell in a grid?

Quick check

Why can two adjacent cells not both be supercells?

Practice Problems

Practice Problems
  1. List five uses of numbers and state what each number tells you.
  2. Find the spoken counts for relative heights 1, 2, 3, 4, 5. Explain what the tallest child says.
  3. Construct a different height order giving 0, 1, 2, 1, 0.
  4. Identify every supercell in 43, 79, 75, 63, 10, 29, 28, 34.
  5. Fill eight cells with distinct numbers so that the maximum number are supercells. Explain your upper bound.
  6. Construct a row where the second smallest is a supercell but the second largest is not.
  7. Check the five-digit construction above in all four directions and record each supercell.
  8. Create a four-digit row whose only supercells are positions 2, 4, and 9 out of nine positions. Use 5346, 1258, and 9635 at positions 1, 4, and 8 respectively.

The counts are 1, 1, 1, 1, 0. Each child except the tallest has a taller child immediately to the right.

Key Takeaways

Key Takeaways

• A number can describe a rule or relationship as well as a quantity. • Count or compare only the neighbours specified by the puzzle. • Supercells are strictly greater than every neighbour; adjacent supercells are impossible. • Alternating high and low values produces the greatest possible number of supercells in a row. • Use examples to show possibility and a reason that covers all arrangements to show impossibility.

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Next · Lesson 2

Number Lines, Place Value, and Digit Sums