Puzzle for March 5, 2020 ( )
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
In eq.5, replace E + F with A (from eq.2): A + B – E = A In the above equation, add E to both sides, and subtract A from both sides: A + B – E + E – A = A + E – A which makes B = E
Hint #2
In eq.3, replace E with B: B – B = A – C which becomes 0 = A – C Add C to each side of the above equation: 0 + C = A – C + C which makes C = A
Hint #3
In eq.4, substitute A for C: A – D = D + E Add D to both sides of the above equation: A – D + D = D + E + D which becomes eq.4a) A = 2×D + E
Hint #4
Substitute 2×D + E for A (from eq.4a) in eq.2: E + F = 2×D + E Subtract E from both sides of the equation above: E + F – E = 2×D + E – E which makes F = 2×D
Hint #5
Substitute E for B, 2×D for F, and A for C in eq.6: E + E – 2×D = A – E + A which becomes 2×E – 2×D = 2×A – E Add E to both sides of the equation above: 2×E – 2×D + E = 2×A – E + E which becomes eq.6a) 3×E – 2×D = 2×A
Hint #6
Substitute (2×D + E) for A (from eq.4a) in eq.6a: 3×E – 2×D = 2×(2×D + E) which is equivalent to 3×E – 2×D = 4×D + 2×E In the above equation, add 2×D to both sides, and subtract 2×E from both sides: 3×E – 2×D + 2×D – 2×E = 4×D + 2×E + 2×D – 2×E which simplifies to E = 6×D and also makes B = E = 6×D
Hint #7
Substitute 6×D for E in eq.4a: A = 2×D + 6×D which makes A = 8×D and also makes C = A = 8×D
Solution
Substitute 8×D for A and C, 6×D for B and E, and 2×D for F in eq.1: 8×D + 6×D + 8×D + D + 6×D + 2×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 = C = 8×D = 8 × 1 = 8 B = E = 6×D = 6 × 1 = 6 F = 2×D = 2 × 1 = 2 and ABCDEF = 868162