Puzzle for December 4, 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
In eq.2, add E to both sides, and subtract B from both sides: B + C + E – B = A – E + E – B which becomes C + E = A – B In eq.5, replace A – B with C + E: C + D = C + E Subtract C from each side: C + D – C = C + E – C which makes D = E
Hint #2
In eq.3, replace E with D: D + D = B which makes 2×D = B
Hint #3
In eq.6, replace E with D: A + D = D + F Subtract D from both sides of the above equation: A + D – D = D + F – D which makes A = F
Hint #4
In eq.5, substitute 2×D for B: C + D = A – 2×D Add 2×D to both sides of the above equation: C + D + 2×D = A – 2×D + 2×D which means C + 3×D = A and also means eq.5a) F = A = C + 3×D
Hint #5
Substitute (C + 3×D) for F (from eq.5a), and 2×D for B in eq.4: C – (C + 3×D) = 2×D – C which is equivalent to C – C – 3×D = 2×D – C which becomes –3×D = 2×D – C Add C and 3×D to both sides of the equation above: –3×D + C + 3×D = 2×D – C + C + 3×D C = 5×D
Hint #6
Substitute 5×D for C in eq.5a: F = A = 5×D + 3×D which makes F = A = 8×D
Solution
Substitute 8×D for A and F, 2×D for B, 5×D for C, and D for E in eq.1: 8×D + 2×D + 5×D + D + D + 8×D = 25 which simplifies to 25×D = 25 Divide both sides of the equation above by 25: 25×D ÷ 25 = 25 ÷ 25 which means D = 1 making A = F = 8×D = 8 × 1 = 8 B = 2×D = 2 × 1 = 2 C = 5×D = 5 × 1 = 5 E = D = 1 and ABCDEF = 825118