Puzzle for May 26, 2021 ( )
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, C, D, and E to both sides of eq.6: B – C + D – E + B + C + D + E = A – B – D + B + C + D + E which simplifies to 2×B + 2×D = A + C + E In the above equation, replace C with A – E (from eq.5): 2×B + 2×D = A + A – E + E which becomes 2×B + 2×D = 2×A Divide both sides by 2: (2×B + 2×D) ÷ 2 = 2×A ÷ 2 which means eq.6a) B + D = A
Hint #2
In eq.6a, replace A with D + F (from eq.2): B + D = D + F Subtract D from each side of the equation above: B + D – D = D + F – D which makes B = F
Hint #3
In eq.3, substitute B for F: E + B = B + C Subtract B from both sides of the equation above: E + B – B = B + C – B which makes E = C
Hint #4
Substitute C for E in eq.5: C = A – C Add C to both sides of the equation above: C + C = A – C + C which makes eq.5a) 2×C = A
Hint #5
Substitute B + D for A (from eq.6a), C for E, and B for F in eq.4: B + D + D + C = B – D + B which becomes B + 2×D + C = 2×B – D In the equation above, subtract B from each side, and add D to each side: B + 2×D + C – B + D = 2×B – D – B + D which becomes eq.4a) 3×D + C = B
Hint #6
Substitute 3×D + C for B (from eq.4a), and 2×C for A in eq.6a: 3×D + C + D = 2×C which becomes 4×D + C = 2×C Subtract C from each side of the above equation: 4×D + C – C = 2×C – C which makes 4×D = C and also makes E = C = 4×D
Hint #7
Substitute 4×D for C in eq.4a: 3×D + 4×D = B which makes 7×D = B and also makes F = B = 7×D
Hint #8
Substitute (4×D) for C in eq.5a: 2×(4×D) = A which makes 8×D = A
Solution
Substitute 8×D for A, 7×D for B and F, and 4×D for C and E in eq.1: 8×D + 7×D + 4×D + D + 4×D + 7×D = 31 which simplifies to 31×D = 31 Divide both sides of the above equation by 31: 31×D ÷ 31 = 31 ÷ 31 which means D = 1 making A = 8×D = 8 × 1 = 8 B = F = 7×D = 7 × 1 = 7 C = E = 4×D = 4 × 1 = 4 and ABCDEF = 874147