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Growing pattern pictures

Patterns and sequences · level 4, lesson 4 of 6

A question to start

A square garden is 5 tiles long on each side, and a gardener wants a border of tiles all the way round its edge (the outer ring of the 5 by 5 square). How many tiles?

Sam says: "Four sides, five tiles each: \(4 \times 5 = 20\)." Priya counts them one by one and gets 16. Who is right, and where did the other one go wrong? This lesson is about seeing a rule inside a picture, and it turns out that seeing it in two different ways is the best check there is.

The idea

Pattern number, picture, table, rule

A growing pattern is a sequence of pictures: pattern 1, pattern 2, pattern 3, ... Each one is built from the one before by a rule. The pattern number plays the part of the position \(n\), and the number of tiles (or dots, or sticks) is the term.

You can find the rule from the table (as in lesson 2: find \(d\), find the zero term). But the picture can give you something better: a reason. If you can see where each part of the rule lives in the picture, you know the rule is right for every pattern number, not just the ones you drew.

L-shapes and crosses

3 5 7 L-shapes: 3, 5, 7, ... 5 9 13 crosses: 5, 9, 13, ...

L-shapes. Pattern \(n\) is an L with two arms. The tiles go 3, 5, 7, ... so \(d = 2\), the zero term is 1, and the rule is \(2n + 1\). Now see it: there is 1 corner tile, plus 2 arms that are each \(n\) tiles long. One corner, two arms of \(n\): \(1 + 2n\). Each new pattern makes both arms 1 longer, which is why the difference is 2.

Crosses. Pattern \(n\) is a plus sign. The tiles go 5, 9, 13, ... so \(d = 4\) and the zero term is 1: the rule is \(4n + 1\). See it: 1 centre tile (blue in the picture) and 4 arms of \(n\) tiles. The zero term 1 is the centre; the 4 in \(4n\) is the number of arms.

This is what makes the picture powerful: the numbers in the rule are things you can point at. The multiplier is "how many parts grow", and the added number is "what stays the same".

The border of a square: seen two ways

Back to the garden. Call the side of the square \(n\). Here is the border of a 5 by 5 square, drawn twice.

4 × 5, corners counted twice: 4 × 5 − 4 = 16 4 strips of 4 (5 − 1): 4 × 4 = 16

Way 1 (left): count the sides, then fix the corners. Each of the 4 sides has \(n\) tiles, which makes \(4n\). But each orange corner tile sits on two sides, so it was counted twice. There are 4 corners, so we counted 4 tiles too many. Border \(= 4n - 4\). For \(n = 5\): \(20 - 4 = 16\).

That is exactly Sam's mistake: his 20 counted every corner twice.

Way 2 (right): cut it into equal strips. Start at a corner and take \(n - 1\) tiles along the top, stopping just before the next corner. Do the same down the right side, along the bottom, and up the left side. Four strips, each \(n - 1\) long, and every tile is in exactly one strip. Border \(= 4(n - 1)\). For \(n = 5\): \(4 \times 4 = 16\).

Two different pictures, two different-looking rules: \(4n - 4\) and \(4(n - 1)\). They must be equal, since they count the same tiles, and they are: 4 lots of \((n - 1)\) is 4 lots of \(n\) take away 4 lots of 1.

More ways. You can also see the border as two full rows of \(n\) (top and bottom) and two short columns of \(n - 2\) (the sides, without the corners already counted): \(2n + 2(n - 2)\), which is \(4n - 4\) again. Or as the whole square take away the inside: the inside is a square of side \(n - 2\), so the border is \(n^2 - (n - 2)^2\). For \(n = 5\): \(25 - 9 = 16\). ✓ Every honest way of seeing gives the same count. When it does not, one of the seeings has a double count or a gap.

Where a rule stops working

Try \(n = 1\): a single tile. The rule \(4n - 4\) says 0, but the "border" of one tile is the tile itself, so the answer should be 1. What broke? Way 1 subtracts 4 corners, but a single tile is all 4 corners at once, so we took away too much. The picture explains when the rule holds: for \(n \ge 2\) (the symbol \(\ge\) means "is greater than or equal to"). A rule always comes with the patterns it is meant for.

Shapes that do not give straight-line rules

Not every growing picture adds the same amount each time. A staircase made of 1, then \(1 + 2\), then \(1 + 2 + 3\) squares (the triangular numbers from level 3) adds 2, then 3, then 4: the difference is not constant, so there is no rule of the form \(dn + c\). Check the differences before you hunt for \(d\) and the zero term.

Worked examples

1. The 20th L-shape. \(2 \times 20 + 1 = 41\) tiles: 1 corner and two arms of 20.

2. The 10th cross. \(4 \times 10 + 1 = 41\) tiles: 1 centre and four arms of 10. (A coincidence: the 20th L and the 10th cross both use 41 tiles. Both have one central tile, and \(2 \times 20 = 4 \times 10\).)

3. The border of a 10 by 10 square, three ways.

  • Sides minus corners: \(4 \times 10 - 4 = 36\).
  • Four strips of 9: \(4 \times 9 = 36\).
  • Whole minus inside: \(100 - 8 \times 8 = 100 - 64 = 36\).

4. Running a rule backwards. Which square has a border of exactly 100 tiles?

  • \(4n - 4 = 100\), so \(4n = 104\) and \(n = 26\).
  • Check: four strips of 25 is 100. ✓

5. Toothpick squares (from level 2), seen. A row of \(n\) squares made of toothpicks uses 4, 7, 10, ... toothpicks: \(3n + 1\). See it: 1 toothpick at the far left, then each square adds 3 more (top, bottom, right side). The "+1" is the starting toothpick; the "3" is the 3 new toothpicks per square.

Try it

  1. A row of triangles made of toothpicks uses 3, 5, 7, ... toothpicks for 1, 2, 3 triangles. Find the rule and say where each number lives in the picture. How many toothpicks for 50 triangles?
  2. Which cross pattern uses 81 tiles?
  3. How many tiles are in the border of a 7 by 7 square? Work it out two ways.
  4. A square's border uses 60 tiles. How long is each side?
  5. Pattern \(n\) is a staircase with columns of 1, 2, ..., \(n\) squares. Is the rule of the form \(dn + c\)? How can you tell?
  6. Explain, with the picture, why \(4n - 4\) gives the wrong answer for a 1 by 1 square.
  7. A "T" pattern has a top bar of 3 tiles and a stem hanging below the middle tile. In pattern \(n\) the stem has \(n\) tiles. Find the rule.
  8. Find the number of tiles in the border of a rectangle 6 tiles by 10 tiles. Then find a rule for the border of a rectangle \(m\) tiles by \(k\) tiles.
Answers
  1. Differences of 2, zero term 1: \(2n + 1\). In the picture: the first triangle's left side is the "1", and each triangle adds 2 more toothpicks (its other two sides). For 50 triangles: \(2 \times 50 + 1 = 101\).
  2. \(4n + 1 = 81\), so \(4n = 80\) and \(n = 20\): pattern 20.
  3. \(4 \times 7 - 4 = 24\); or four strips of 6, \(4 \times 6 = 24\); or \(49 - 25 = 24\).
  4. \(4n - 4 = 60\), \(4n = 64\), \(n = 16\). Check: four strips of 15 is 60. ✓
  5. No. The totals are 1, 3, 6, 10, ... with differences 2, 3, 4, ... which are not constant.
  6. Way 1 counts 4 sides of 1 tile (4 tiles) and then subtracts 4 corners. But the single tile is all four corners at once: it was counted 4 times, so only 3 should be taken away, giving 1. The rule was built for squares with separate corners, which means \(n \ge 2\).
  7. 3 tiles in the bar and \(n\) in the stem: \(n + 3\). (Table: 4, 5, 6, ... difference 1, zero term 3 ✓.)
  8. Sides: \(6 + 10 + 6 + 10 = 32\), minus 4 double-counted corners: 28. (Check: whole \(60\) minus inside \(4 \times 8 = 32\) gives 28 ✓.) In general: \(2m + 2k - 4\).

Common mistake

Counting the corners twice. Adding up the sides of a border, a frame or a fence counts every corner tile on two sides. Ask "is any tile in two of my groups?" and either subtract the repeats or cut the shape into pieces that do not overlap.

Trusting the table and never looking at the picture. The table gives a rule; the picture tells you why it is right and for which pattern numbers. If your rule's numbers do not match anything you can point at, look again.

Where it leads