Puzzle for October 24, 2019 ( )
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.6, replace B with D + E (from eq.2): D + E – C = D – E Subtract D from each side of the above equation: D + E – C – D = D – E – D which becomes E – C = –E Add C and E to both sides: E – C + C + E = –E + C + E which makes 2×E = C
Hint #2
In eq.3, replace B with D + E (from eq.2): E + F = A + D + E Subtract E from both sides of the above equation: E + F – E = A + D + E – E which becomes eq.3a) F = A + D
Hint #3
In eq.5, replace F with A + D (from eq.3a): A + D = C + D Subtract D from both sides of the above equation: A + D – D = C + D – D which makes A = C and which also makes A = C = 2×E
Hint #4
In eq.4, substitute C + D for F (from eq.5): B + D = A + C + C + D which becomes B + D = A + 2×C + D Subtract D from both sides of the above equation: B + D – D = A + 2×C + D – D which becomes eq.4a) B = A + 2×C
Hint #5
Substitute 2×E for both A and C in eq.4a: B = 2×E + 2×2×E which is the same as B = 2×E + 4×E which means B = 6×E
Hint #6
Substitute 6×E for B in eq.2: D + E = 6×E Subtract E from both sides: D + E – E = 6×E – E which makes D = 5×E
Hint #7
Substitute 2×E for C, and 5×E for D in eq.5: F = 2×E + 5×E which makes F = 7×E
Solution
Substitute 2×E for A and C, 6×E for B, 5×E for D, and 7×E for F in eq.1: 2×E + 6×E + 2×E + 5×E + E + 7×E = 23 which simplifies to 23×E = 23 Divide both sides of the above equation by 23: 23×E ÷ 23 = 23 ÷ 23 which means E = 1 making A = C = 2×E = 2 × 1 = 2 B = 6×E = 6 × 1 = 6 D = 5×E = 5 × 1 = 5 F = 7×E = 7 × 1 = 7 and ABCDEF = 262517