Puzzle for September 6, 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 + C with A + D + E (from eq.2): A + D + E + E = A + D + F which is equivalent to A + D + 2×E = A + D + F Subtract both A and D from each side of the above equation: A + D + 2×E – A – D = A + D + F – A – D which simplifies to 2×E = F
Hint #2
Add D and E to both sides of eq.5: A – E + D + E = B – D + D + E which becomes A + D = B + E In eq.2, replace A + D with B + E: B + C = B + E + E Subtract B from each side of the equation above: B + C – B = B + E + E – B which makes C = 2×E and also makes eq.2a) F = C = 2×E
Hint #3
In eq.3, substitute 2×E for C and F: 2×E + E + 2×E = B + D which becomes 5×E = B + D Subtract D from both sides of the above equation: 5×E – D = B + D – D which becomes eq.3a) 5×E – D = B
Hint #4
In eq.4, substitute 2×E for C: D = A + 2×E Subtract 2×E from each side of the equation above: D – 2×E = A + 2×E – 2×E which becomes eq.4a) D – 2×E = A
Hint #5
Substitute D – 2×E for A (from eq.4a), and 5×E – D for B (from eq.3a) in eq.5: D – 2×E – E = 5×E – D – D which becomes D – 3×E = 5×E – 2×D Add 3×E and 2×D to each side of the above equation: D – 3×E + 3×E + 2×D = 5×E – 2×D + 3×E + 2×D which becomes 3×D = 8×E Divide both sides by 8: 3×D ÷ 8 = 8×E ÷ 8 which makes ⅜×D = E
Hint #6
Substitute (⅜×D) for E in eq.2a: F = C = 2×(⅜×D) which makes F = C = ¾×D
Hint #7
Substitute (⅜×D) for E in eq.4a: D – 2×(⅜×D) = A which becomes D – ¾×D = A which makes ¼×D = A
Hint #8
Substitute (⅜×D) for E in eq.3a: 5×(⅜×D) – D = B which becomes 1⅞×D – D = B which makes ⅞×D = B
Solution
Substitute ¼×D for A, ⅞×D for B, ¾×D for C and F, and ⅜×D for E in eq.1: ¼×D + ⅞×D + ¾×D + D + ⅜×D + ¾×D = 32 which simplifies to 4×D = 32 Divide both sides of the equation above by 4: 4×D ÷ 4 = 32 ÷ 4 which means D = 8 making A = ¼×D = ¼ × 8 = 2 B = ⅞×D = ⅞ × 8 = 7 C = F = ¾×D = ¾ × 8 = 6 E = ⅜×D = ⅜ × 8 = 3 and ABCDEF = 276836