Patterns in Mathematics: questions with answers

The chart forgot the dot in the centre. A hexagonal number is 6 times a triangular number and then 1 more for the centre dot, so the next one is $6 \times 15 + 1 = 91$, not 90. You can check it with the rings: the rings hold 6, 12, 18, ... dots, so the new ring around 61 holds 30 dots, and $61 + 30 = 91$.

Multiply each triangular number by 6, then add 1 for the centre dot. This gives 1 × 6 + 1 = 7, 3 × 6 + 1 = 19, and 6 × 6 + 1 = 37.

Start at 1 and multiply by 3 each time: 1 × 3 = 3, 3 × 3 = 9, 9 × 3 = 27. The first four numbers are 1, 3, 9, 27. The list 3, 6, 9, 12 adds 3 each time and gives multiples of 3.

They added instead of multiplying. 2 + 2 + 2 = 6, but three 2s multiplied together make 2 × 2 × 2 = 8. Adding 2 again and again gives a multiple of 2. Multiplying by 2 again and again gives a power of 2, and powers grow much faster. 9 comes from multiplying 3 by itself, and 12 comes from 2 × 2 × 3. Neither of those is three 2s multiplied together.

No. Her rule gives 5, 8, 11 at the third term, and the list says 12. A rule has to hold for every term shown, not just the first step -- the first step being 3 is what makes the mistake so easy to make. Replacing it with the last step, 5, repeats exactly the same error at the other end of the list; that rule fails at the very first term. Four terms are quite enough to settle it: what the steps do between them is already visible.

26. The steps between the terms are 3, 5 and 7. Each step is 2 more than the one before. So the next step is 9, and the next term is 17 + 9 = 26. Repeating the last step of 7 gives 24. That forgets that the steps keep growing. The answer 9 is the size of the next step, not the next term. The steps make a list of their own, but it is a different list from this one. Doubling 17 gives 34, and this list has never followed that rule.

Add 6 dots along one side, 6 along another side, and 1 at the new corner. This adds 6 + 6 + 1 = 13 dots. The next square has 49 dots in all, so we can check: 49 - 36 = 13.

Adding the next ring adds 12 + 6 = 18 dots. The whole figure then has 19 + 18 = 37 dots.

The bands contain 1, 3, 5, 7, 9, 11, 13, 15 dots, from the corner outwards. Each number counts the dots in one band. There are 8 numbers, so there are 8 bands. Starting at 3 would leave out the corner dot's band.

49. The first 7 odd numbers are 1, 3, 5 and so on up to 13 -- they begin at 1, because 1 is odd -- and adding them gives 7 x 7 = 49. Starting instead at 3 would give 3, 5, 7 and so on up to 15, which comes to 63: a different sum altogether, and one short of a square. Multiplying 7 by the last term 13 gives 91 and assumes every term is as big as the last, when they run all the way down to 1.

144. A climb to a peak and back adds to the peak multiplied by itself -- the chapter reaches 10,000 for a peak of 100 -- so this one comes to 12 x 12 = 144. Doubling the peak gives 24, which would be the total if the class said only 12 twice and nothing else. Counting how many numbers were said gives 23, which answers a different question: 23 numbers were spoken, and they add to 144.

16 small triangles. Count the triangles pointing up and those pointing down between them. The 4 rows hold 1, 3, 5 and 7 small triangles. Add the rows: 1 + 3 + 5 + 7 = 16. That is 4 x 4, a square number. Counting only as many triangles as the row number would give 10. That counts dots in a dot triangle, not small triangles in this shape. Squaring the last row count, 7, gives 49. Multiplying the 4 rows by the last row count gives 28. But the rows do not all hold 7 small triangles. It is the number of rows, 4, that gets squared.

6 lines. A closed shape has as many corners as it has sides. Going round it, you meet a side, then a corner, then a side again, and you end where you started. So the square has 4 corners, not 3. Now join the corners one at a time: the 2nd corner adds 1 line, the 3rd adds 2 more, and so on, until the last corner adds 3. That gives $1 + 2 + 3 = 6$, a triangular number, just like the shape with 4 points in the Complete Graphs sequence. The answer is not 4 lines, because each corner joins to all 3 other corners, not just one.

A speed bump turns one line segment into 4 smaller ones: the two flat pieces on either side, and the two sides of the bump. So each stage has 4 times as many line segments as the stage before it. The counts go 3, 12, 48, 192, 768, and stage 2 has $3 \times 4 = 12$. If you count a bump as only 3 segments, you get 9. If you multiply the stage number by 3, you get 6, but the 3 is only the triangle you start with. 3 is the count for the stage before, so it stops one step too early.

768. A speed bump leaves four segments where there was one line segment. So the count is multiplied by 4 at each stage. That makes 192 become 4 × 192 = 768 after one stage. Adding a fixed nine gives 201. That rule fits only the jump from 3 to 12. Three times the stage number gives 15. The chapter says the counts are 3 times Powers of 4. It is the Powers of 4 that grow, and the 3 is only the triangle at the start.