Puzzle for August 4, 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.4, replace F with B + C (from eq.2): C + D = B + C – D In the equation above, subtract C from both sides, and add D to both sides: C + D – C + D = B + C – D – C + D which becomes 2×D = B
Hint #2
In eq.6, add F to both sides, and subtract C from both sides: A – F + F – C = B + C + D + F – C which becomes A – C = B + D + F In eq.3, replace B + D + F with A – C: A – C = A – B Subtract A from each side of the above equation: A – C – A = A – B – A which makes –C = –B which means C = B and also means C = B = 2×D
Hint #3
In eq.2, substitute 2×D for B and C: F = 2×D + 2×D which makes F = 4×D
Hint #4
Substitute 2×D for B, and 4×D for F in eq.3: 2×D + D + 4×D = A – 2×D which becomes 7×D = A – 2×D Add 2×D to both sides of the equation above: 7×D + 2×D = A – 2×D + 2×D which makes 9×D = A
Hint #5
Substitute 2×D for B and C, and 9×D for A in eq.5: 2×D – 2×D = 9×D – E which becomes 0 = 9×D – E Add E to each side of the equation above: 0 + E = 9×D – E + E which makes E = 9×D
Solution
Substitute 9×D for A and E, 2×D for B and C, and 4×D for F in eq.1: 9×D + 2×D + 2×D + D + 9×D + 4×D = 27 which simplifies to 27×D = 27 Divide both sides of the above equation by 27: 27×D ÷ 27 = 27 ÷ 27 which means D = 1 making A = E = 9×D = 9 × 1 = 9 B = C = 2×D = 2 × 1 = 2 F = 4×D = 4 × 1 = 4 and ABCDEF = 922194