Puzzle for July 20, 2023 ( )
Scratchpad
Find the 6-digit number ABCDEF by solving the following equations:
A, B, C, D, E, and F each represent a one-digit non-negative integer.
Scratchpad
Help Area
Hint #1
In eq.3, replace D with E + F (from eq.2): A + E = E + F + F which becomes A + E = E + 2×F Subtract E from each side of the equation above: A + E - E = E + 2×F - E which makes A = 2×F
Hint #2
In eq.4, replace A with 2×F: C - E = 2×F + F which becomes C - E = 3×F Add E to both sides of the above equation: C - E + E = 3×F + E which becomes eq.4a) C = 3×F + E
Hint #3
In eq.5, substitute 3×F + E for C (from eq.4a): B - E = 3×F + E + F which becomes B - E = 4×F + E Add E to both sides of the equation above: B - E + E = 4×F + E + E which becomes eq.5a) B = 4×F + 2×E
Hint #4
Substitute (3×F + E) for C (from eq.4a), 2×F for A, and 4×F + 2×E for B (from eq.5a) in eq.6: (3×F + E) + F = 2×F + 4×F + 2×E - (3×F + E) + E which becomes 4×F + E = 6×F + 3×E - 3×F - E which becomes 4×F + E = 3×F + 2×E Subtract E and 3×F from each side of the above equation: 4×F + E - E - 3×F = 3×F + 2×E - E - 3×F which simplifies to F = E
Hint #5
Substitute F for E in eq.5a: B = 4×F + 2×F which makes B = 6×F
Hint #6
Substitute F for E in eq.4a: C = 3×F + F which makes C = 4×F
Hint #7
Substitute F for E in eq.2: D = F + F which makes D = 2×F
Solution
Substitute 2×F for A and D, 6×F for B, 4×F for C, and F for E in eq.1: 2×F + 6×F + 4×F + 2×F + F + F = 16 which simplifies to 16×F = 16 Divide both sides of the above equation by 16: 16×F ÷ 16 = 16 ÷ 16 which means F = 1 making A = D = 2×F = 2 × 1 = 2 B = 6×F = 6 × 1 = 6 C = 4×F = 4 × 1 = 4 E = F = 1 and ABCDEF = 264211