Patterns in Mathematics · Lesson 4 of 7
Surprising Links Between Sequences
“Discover how doubling, triangular groups, hexagonal pictures, and cubes are connected.”
• Explain why a running total of powers of 2 is one less than the next power. • Build a hexagonal-number picture from six triangular groups and a centre. • Connect running totals of the hexagonal sequence to cube numbers. • Use a picture and an arithmetic check to justify a proposed relationship. • Develop and explain a new pattern within the chapter’s sequences.
A doubling total is almost the next term
Adding the powers of 2 gives 1; 1 + 2 = 3; 1 + 2 + 4 = 7; and 1 + 2 + 4 + 8 = 15. These totals are close to powers of 2. Add one to each total and you get 2, 4, 8, and 16, which are the next terms of the doubling sequence.
| Last term included | Running total | After adding 1 |
|---|---|---|
| 1 | 1 | 2 |
| 2 | 1 + 2 = 3 | 4 |
| 4 | 1 + 2 + 4 = 7 | 8 |
| 8 | 1 + 2 + 4 + 8 = 15 | 16 |
| 16 | 1 + 2 + 4 + 8 + 16 = 31 | 32 |
A useful picture starts with one extra counter. Place it beside the first 1 from the sequence: together they make 2. Join that group with the sequence’s 2 to make 4. Join the new group with the sequence’s 4 to make 8. Each time the group already built is exactly the size of the next group being added, so the combined group doubles.
This explains why the relationship continues. Whenever the total plus one matches the next power of 2, adding that next power doubles the completed group again. The running total itself remains one less than the new completed group. We can express the result in words: add successive powers of 2 starting at 1, and the answer is one less than the next power of 2.
Problem
Find 1 + 2 + 4 + 8 + 16 + 32.
- 1.The final included term is 32, so the next power of 2 is 32 × 2 = 64.
- 2.The running total plus one is 64.
- 3.Therefore the requested sum is 64 − 1 = 63. A check by grouping gives 1 + 2 + 4 + 8 + 16 = 31, then 31 + 32 = 63.
The total is one less than the next power, not one less than the last term already included. Also, this particular rule depends on starting at 1 and including every successive power of 2.
Six triangles around one centre
Recall the hexagonal dots 1, 7, 19, 37, 61, …. Separate the centre dot from the surrounding dots. The remaining dots can be divided into six equal triangular groups. As the hexagon grows, each group changes from 1 to 3 to 6 to 10 dots: the triangular-number sequence.
The 19-dot picture has six groups of 3 and one centre dot. The 37-dot picture has six groups of 6 and one centre dot. This is the meaning of multiplying a triangular number by 6 and adding 1. All six groups are needed, and the centre belongs to none of them, so it is added once.
| Triangular group size | Six groups and a centre | Hexagonal total |
|---|---|---|
| 1 | 6 × 1 + 1 | 7 |
| 3 | 6 × 3 + 1 | 19 |
| 6 | 6 × 6 + 1 | 37 |
| 10 | 6 × 10 + 1 | 61 |
| 15 | 6 × 15 + 1 | 91 |
The initial hexagonal number 1 is just the centre before any surrounding groups have been added. Thus multiplying the positive triangular sequence by 6 and adding 1 produces the hexagonal sequence starting at 7. Keep the initial 1 when you later add up the full hexagonal sequence.
Problem
The next triangular group has 21 dots. How many dots are in the corresponding hexagonal picture?
- 1.There are six copies of the 21-dot group, making 6 × 21 = 126 surrounding dots.
- 2.Add the single centre: 126 + 1 = 127 dots.
- 3.This also follows the ring rule: after 91, the next outer ring adds 36, giving 91 + 36 = 127.
Hexagonal totals build cubes
Now form running totals of the full hexagonal sequence: 1; 1 + 7 = 8; 1 + 7 + 19 = 27; and 1 + 7 + 19 + 37 = 64. These are cube numbers. To explain the connection, imagine growing a solid cube one unit longer in each direction. The added blocks form a corner-shaped outer layer.
Begin with a 2-by-2-by-2 cube containing 8 unit blocks. To make a 3-by-3-by-3 cube, add a full top layer of 9 blocks. Below that layer, add a front strip with 2 rows of 3 blocks, making 6. Then add the remaining side strip with 2 rows of 2 blocks, making 4. The three pieces meet but do not overlap, so they add 9 + 6 + 4 = 19 blocks.
The same construction explains later stages. Growing a 3-by-3-by-3 cube into a 4-by-4-by-4 cube adds 16 + 12 + 9 = 37 blocks. Those are the next hexagonal dots. One can view each corner layer along the cube’s long diagonal and arrange its blocks into the six triangular groups around a centre.
For a general side length n, the three added pieces have n × n, n × (n − 1), and (n − 1) × (n − 1) blocks. Here n is the new side length, and n − 1 is the old side length. Their combined count matches the hexagonal arrangement with a centre and six triangular groups through row n − 1. The pictures therefore link a flat dot pattern to the blocks needed at each step of a growing cube.
We can check that match using our earlier triangular relationships. Call the triangular number through row n − 1 “T”. The top n-by-n layer is two consecutive triangular groups: T and the next triangular number, T + n. The front n-by-(n − 1) strip splits into two T-sized triangles. The side (n − 1)-by-(n − 1) strip splits into T and the preceding triangular number, T − (n − 1). Altogether there are six copies of T, with an extra n and a subtraction of n − 1. Those extras leave exactly one block. The corner layer therefore contains 6 × T + 1 blocks, matching the hexagonal count. For the first growth from side 1 to side 2, the same grouping uses an empty preceding triangle of zero dots.
Problem
Find 1 + 7 + 19 + 37 + 61.
- 1.These are the first five numbers of the chapter’s hexagonal sequence, including the initial centre-only term 1.
- 2.Each is the addition that grows a cube to the next side length: 1, then 2, then 3, then 4, then 5.
- 3.After five stages, the total fills a 5-by-5-by-5 cube, so it is 5 × 5 × 5 = 125. Check: 1 + 7 = 8; +19 = 27; +37 = 64; +61 = 125.
A hexagonal number such as 19 is the amount added at one cube-building stage. It is not the cube’s total. The cube total after that stage is 1 + 7 + 19 = 27.
Discover a relationship and defend it
A pattern investigation has two parts. First, make a clear observation from several terms or pictures. Then explain the observation using how the objects are built. A calculation can check your claim in a particular case; a construction can show why it should work in every stage covered by the rule.
Problem
What happens when an odd number is increased by 1?
- 1.Try 1 + 1 = 2, 3 + 1 = 4, 5 + 1 = 6, and 7 + 1 = 8. The answers are successive even numbers.
- 2.An odd dot group has pairs plus one leftover dot. The new dot pairs with that leftover dot.
- 3.The result contains only pairs, so it is even. The picture explains the relationship for any odd number.
Choose sequences from the chapter. Compare terms, running totals, or the amount added between pictures. State your conjecture—a claim you think may be true—then check it in several cases and seek a dot, grid, or cube explanation. If you cannot yet explain it, describe exactly what remains to be justified.
Quiz
What is 1 + 2 + 4 + 8 + 16?
What must be added to that running total to reach the next power of 2?
Six triangular groups of 10 dots and one centre contain:
Which term is missing if 6 times the positive triangular numbers plus 1 gives 7, 19, 37, …?
What is 1 + 7 + 19 + 37?
What does the 19 represent when a 2-by-2-by-2 cube grows into a 3-by-3-by-3 cube?
Practice Problems
- Find 1 + 2 + 4 + 8 + 16 + 32 + 64 using the next power of 2. Explain why your method works.
- Use one extra counter to explain why 1 + 2 + 4 + 8 is one less than 16.
- Draw 19 hexagonal dots as six triangular groups of 3 and one centre. Then describe the corresponding grouping for 37.
- Use triangular numbers 10 and 15 to obtain two hexagonal totals. Check them using the ring-growth rule.
- Find the sum of the first six hexagonal numbers 1, 7, 19, 37, 61, 91. Explain its cube interpretation.
- How many unit blocks are added when a 3-by-3-by-3 cube becomes a 4-by-4-by-4 cube? Count a top layer and two non-overlapping strips.
- A student claims 1 + 7 + 19 equals 19 because 19 is the last hexagonal number. Explain the distinction between the final addition and the running total.
- Find a new relation among the chapter’s sequences. Give three numerical checks and a picture or explanation for why it occurs.
Key Takeaways
• Running totals of powers of 2 beginning at 1 are one less than the next power. • Six equal triangular groups plus one centre give the hexagonal numbers after the initial 1. • Running totals of 1, 7, 19, 37, … are cube numbers. • A cube’s outer corner layer explains the difference between successive cube totals. • State a pattern, check it, and explain the construction that makes it work.