Step by Step solution about How to resolve the algorithm Pascal's triangle/Puzzle step by step in the Mathematica/Wolfram Language programming language

This Wolfram code solves a system of linear equations to find the values of a set of unknown variables. The equations represent the sums of adjacent elements in a triangular array.

The first block of code defines the equations and the known values:

• b+c==a
• d+e==b
• e+f==c
• g+h==d
• h+i==e
• i+j==f
• l+X==g
• l+Y==h
• n+Y==i
• n+Z==j
• X+Z==Y

The known values are:

• a=151
• d=40
• l=11
• n=4

The Solve function is then used to find the values of the unknown variables, b, c, e, f, g, h, i, j, X, Y, and Z. The solution is:

{{b -> 81, c -> 70, e -> 41, f -> 29, g -> 16, h -> 24, i -> 17, j -> 12, X -> 5, Y -> 13, Z -> 8}}

This solution can be represented as a triangular array:

151
81 70
40 41 29
16 24 17 12
5 11 13 4 8

The second block of code uses the triangle function to generate a triangular array with a given number of rows and columns. The Nest function is used to repeatedly apply the MovingMap function to the array, which adds the values of adjacent elements in each row. The Solve function is then used to find the values of the variables x, y, and z that satisfy the given equations:

• triangle[3, 1] == 40
• triangle[5, 1] == 151
• y == x + z

The solution is:

{{x -> 5, y -> 13, z -> 8}}

b+c==a
d+e==b
e+f==c
g+h==d
h+i==e
i+j==f
l+X==g
l+Y==h
n+Y==i
n+Z==j
X+Z==Y


a->151
d->40
l->11
n->4


eqs={a==b+c,d+e==b,e+f==c,g+h==d,h+i==e,i+j==f,l+X==g,l+Y==h,n+Y==i,n+Z==j,Y==X+Z};
knowns={a->151,d->40,l->11,n->4};
Solve[eqs/.knowns,{b,c,e,f,g,h,i,j,X,Y,Z}]


{{b -> 81, c -> 70, e -> 41, f -> 29, g -> 16, h -> 24, i -> 17,  j -> 12, X -> 5, Y -> 13, Z -> 8}}


				151
			81		70
		40		41		29
	16		24		17		12
5		11		13		4		8


triangle[n_, m_] :=  Nest[MovingMap[Total, #, 1] &, {x, 11, y, 4, z}, n - 1][[m]]
Solve[{triangle[3, 1] == 40, triangle[5, 1] == 151, y == x + z}, {x, y, z}]


{{x -> 5, y -> 13, z -> 8}}