Puzzle for November 13, 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
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Hint #1
Add B to each side of eq.6: C – B + B = B + D + B which becomes eq.6a) C = 2×B + D In eq.5, replace C with 2×B + D (from eq.6a): B + F = A – B + 2×B + D which becomes B + F = A + B + D Subtract B from both sides of the equation above: B + F – B = A + B + D – B which becomes eq.5a) F = A + D
Hint #2
In eq.2, replace A + D with F (from eq.5a): eq.2a) C = F + E
Hint #3
In eq.4, substitute F + E for C (from eq.2a): F = F + E – D In the equation above, subtract F from each side, and add D to each side: F – F + D = F + E – D – F + D which makes D = E
Hint #4
Substitute A + D + E for C (from eq.2) in eq.3: A + B = A + D + E + D + E which becomes A + B = A + 2×D + 2×E Subtract A from both sides of the above equation: A + B – A = A + 2×D + 2×E – A which becomes eq.3a) B = 2×D + 2×E
Hint #5
Substitute D for E in eq.3a: B = 2×D + 2×D which makes B = 4×D
Hint #6
Substitute (4×D) for B in eq.6a: C = 2×(4×D) + D which becomes C = 8×D + D which means C = 9×D
Hint #7
Substitute 9×D for C, and D for E in eq.2a: 9×D = F + D Subtract D from each side of the above equation: 9×D – D = F + D – D which makes 8×D = F
Hint #8
Substitute 8×D for F in eq.5a: 8×D = A + D Subtract D from each side of the equation above: 8×D – D = A + D – D which makes 7×D = A
Solution
Substitute 7×D for A, 4×D for B, 9×D for C, D for E, and 8×D for F in eq.1: 7×D + 4×D + 9×D + D + D + 8×D = 30 which simplifies to 30×D = 30 Divide both sides of the equation above by 30: 30×D ÷ 30 = 30 ÷ 30 which means D = 1 making A = 7×D = 7 × 1 = 7 B = 4×D = 4 × 1 = 4 C = 9×D = 9 × 1 = 9 E = D = 1 F = 8×D = 8 × 1 = 8 and ABCDEF = 749118