38 Azim Premji University At Right Angles, November 2022
Problem Corner
While working on a counting problem, I encountered the need to compute the sum
(5)(1) + (4)(2) + (3)(3) + (2)(4) + (1)(5) and I was immediately reminded of Pascal’s triangle and the Rascal Triangle.a I took a detour and looked at a relative of the Rascal Triangle, shown in Figure 1.
(1)(1)
(2)(1) (1)(2)
(3)(1) (2)(2) (1)(3)
(4)(1) (3)(2) (2)(3) (1)(4)
(5)(1) (4)(2) (3)(3) (2)(4) (1)(5)
(m)(1) (1)(m)
1
2 2
3 4 3
4 6 6 4
5 8 9 8 5
6 10 12 12 10 6
Figure 1
a I needed to find the sum of 1+ (1+2) + (1+2+3) + (1+2+3+4)+
(1+2+3+4+5). If you count the number of 1’s in this expression, the number of 2’s, the number of 3’s, …, you get the expression shown above.
1
Keywords: Number patterns, sums, Rascal Triangle
A Cousin of the Rascal Triangle
JAMES METZ
Azim Premji University At Right Angles, November 2022 39
The first row is row 0. Along each diagonal is a sequence of numbers with a common difference, first of 1, then 2, then 3, etc. Also, the central vertical column consists of the squares of the integers. The central number in row n (for n even) is (n+42)2 = n2
4 +n+1.
The sum of the numbers in row n is given by
(n+1) (n+2) (n+3)
6 , which is the product of three consecutive integers, divided by 6. For example, the sum of the numbers in row 5 is
(6) (7) (8)
6 = 56.
Suppose we want to know the 3rdterm in row 5, as shown in Figure 2, with the diamond showing a=6, b=8, c=9 and x.
The value of x can be found using the formula x=b+c+1−a, so x=8+9+1−6=12.
This is the same formula used with the Rascal Triangle. We cannot use the formula x= (bc+1)/a, the second formula for a Rascal Triangle, but instead we can use x=bc/a, noting that the product of the North and South numbers equals the product of the East and West numbers.
Note that the sum of the North and South numbers and the sum of the East and West numbers differ by 1.
2 1 2 3 4 3 4 6 6 4 5 8 9 8 5
x Figure 2
The value of x can also be found using the equation x=a+n+1, which is different from the Rascal Triangle equation x=a+n−1. For example, with a=6 and n=5, x =6+
5+1=12. For another example, if we want to find the 2ndelement in row 8, a=7 by
inspection, and n=8, so x=7+8+1=16.
Substituting a=b+c+1−x for a in x=a+ n+1 and solving for x, we have,
x= (b+c+n+2)/2,
an equation that requires only b, c and the row number n, not unlike what was done with Pascal’s Triangle but also using n. See Figure 3. To find the 3rdterm in row 5, x =(8+9+5+2)/2=12.
2 1 2 3 4 3 4 6 6 4 5 8 9 8 5
x
Figure 3
I never would have considered this investigation had I not known about the Rascal Triangle, so I thank the authors of “The Rascal Triangle.”
References
1. The Rascal Triangle from the March 2016 issue of At Right Angles available on
http://publications.azimpremjifoundation.org/3003/1/AtRia_Rascal%20Triangle_Mar%202016.pdf
JAMES METZ [email protected], is a retired mathematics instructor. He volunteers for Teachers Across Borders - Southern Africa for one month each year and enjoys doing mathematics.
