At the beginning of the meeting, we shared some favorite sources of puzzles we like to use with students, include Which One Doesn’t Belong, Sometimes, Always, Never, and Open Middle.
Then I introduced Number Pyramids. Thank you to Henri Piccioto and his amazing web site of math resources. Here is the sequence we used:
Try these pyramids on your own.
Some shared language
2-row pyramid, 3-row pyramid, 4-row pyramid, …
seeds: the numbers that are given
middle number/cell: a number or cell that does not have an outside edge
wings: We decided on this term to refer to the bottom leftmost and bottom rightmost numbers
What do you notice?
- Adding two even numbers yields an even sum
- Adding an odd and an even number yields an odd sum
- The pyramids on the bottom – the numbers given are all on the outside diagonal
- In 4-row pyramids, the sums of each row increase by the same constant number found in the middle cell of 2nd row, or middle two cells of bottom row.
- The sum of the top of a 3-row pyramid is double the center bottom number, plus the wings (bottom corner numbers).
- The sum of the top of a 4-row is triple the middle number, plus the wings.
- Two matching numbers lead to a “mirror pattern” in the row below: 7, 7 leads to 3, 4, 3 below.
Mark noticed that the sum of each row a 3-row pyramid and a 4-row pyramid increase by a constant value that can be found in the middle of the bottom row of a 3-row pyramid and in the middle cell of the 4-row pyramid.
Maggie noticed that in a 3-row pyramid, the top number is 2 times the bottom middle number, plus the wings. In a 4-row pyramid, the top number is 3 times the middle number, plus the wings.
What do you wonder?
- Would a 5-row pyramid have Maggie’s pattern? Conjecture: 4 times all the sums of the middle bottom numbers, plus the wings.
- Is this only a whole number puzzle? Could the boxes have decimals or fractions? Could they have negative numbers?
- What’s the fewest seeds needed to solve a puzzle? Where do the seeds need to be in the pyramid?
- Given the top number, can we fill in any-sized pyramid?
- Could we fill in a blank pyramid? Could that be a puzzle?
- Do we need numbers at all?
- At what point does it become too difficult to solve? 2 seeds on a 5-row pyramid, for example. Would we still be able to solve it?
- How do you choose the seeds and the positions of the numbers? Do only 2-digit numbers work? Is it possible with 3-digit and 4-digit numbers?
- Is there an application for this pattern? Building a building, road, etc?
- Is there a certain number of seeds where a finite number of solutions is possible? Fewer seeds where multiple solutions are possible?
- How could we represent this algebraically? How could we make the unfolding of Mark’s and Maggie’s patterns explicit?
- What would happen if we changed the pyramid rule? Subtraction, multiplication, …
Small Groups
We then spent about 15 minutes working individually to consider the questions we generated, then worked in groups.
Nadia and Amy experimented with making number pyramid puzzles with fractions.
Rebecca, Older, and Mark examined the pattern of the constant growth in the sum of the pyramid rows and found that the pattern changed in a 5-row pyramid. In the pyramids below, there is a constant difference between rows 2 and 3, and rows 3 and 4, but the difference changes between rows 4 and 5.
Maggie, Sarah, and Spencer used algebra to analyze a 6-row pyramid.
They found patterns in the coefficients of the variables for the numbers used in the base row: A, B, C, D, E, F. They noticed that the coefficients of the variables within each of the cells are the same as a row in Pascal’s Triangle. The coefficients of the variables in the top cell is also the same as a row in Pascal’s Triangle.
- The coefficients for the total of row 1 are 1, 1, 1, 1, 1, 1.
- The coefficients for the total of row 2 are 1, 2, 2, 2, 2, 1.
- The coefficients for the total of row 3 are 1, 3, 4, 4, 3, 1.
- The coefficients for the total of row 4 are 1, 4, 7, 7, 4, 1.
- The coefficients for the total of row 5 are 1, 5, 10, 10, 5, 1.
Additional puzzles/questions to consider
- Knowing the three corner numbers in a 3-row pyramid, is it possible to predict the number that will go in the middle of the bottom row without any trial and error?
- Knowing the three corner numbers in a 4-row pyramid, is it possible to predict the number that will go in the middle cell without any trial and error?
- How many seeds are necessary to create one solution? Does the position change how many seeds are necessary?
- Make a pyramid puzzle that cannot be solved. Make a pyramid puzzle that can be solved in more than one way.
- What would be a useful sequence of puzzles for a group of students?
If you create more number pyramids, please share them so that we can add them to our worksheet for students.
