Puzzle for March 21, 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
Add F to each side of eq.6: E – F + F = C + F + F which becomes E = C + 2×F In the above equation, replace E with C + D (from eq.2): C + D = C + 2×F Subtract C from both sides: C + D – C = C + 2×F – C which makes D = 2×F
Hint #2
In eq.3, replace E with C + D (from eq.2): D + C + D + F = A + B + C which becomes C + 2×D + F = A + B + C Subtract C from both sides of the equation above: C + 2×D + F – C = A + B + C – C which becomes eq.3a) 2×D + F = A + B
Hint #3
Subtract the left and right sides of eq.4 from the left and right sides of eq.3a, respectively: 2×D + F – (A + F) = A + B – (B + D) which is the same as 2×D + F – A – F = A + B – B – D which becomes 2×D – A = A – D Add A and D to each side of the above equation: 2×D – A + A + D = A – D + A + D which becomes 3×D = 2×A Divide both sides by 2: 3×D ÷ 2 = 2×A ÷ 2 which makes eq.3b) 1½×D = A
Hint #4
In eq.4, substitute 1½×D for A: 1½×D + F = B + D Subtract D from both sides of the equation above: 1½×D + F – D = B + D – D which becomes eq.4a) ½×D + F = B
Hint #5
Substitute (2×F) for D in eq.4a: ½×(2×F) + F = B which becomes F + F = B which makes 2×F = B
Hint #6
Substitute (2×F) for D in eq.3b: 1½×(2×F) = A which makes 3×F = A
Hint #7
Substitute 2×F for D in eq.2: eq.2a) C + 2×F = E
Hint #8
Substitute 3×F for A, and (C + 2×F) for E (from eq.2a) in eq.5: 3×F – C – (C + 2×F) = C which is the same as 3×F – C – C – 2×F = C which becomes F – 2×C = C Add 2×C to both sides of the above equation: F – 2×C + 2×C = C + 2×C which becomes F = 3×C Divide both sides by 3: F ÷ 3 = 3×C ÷ 3 which makes ⅓×F = C
Hint #9
Substitute ⅓×F for C in eq.2a: ⅓×F + 2×F = E which makes 2⅓×F = E
Solution
Substitute 3×F for A, 2×F for B and D, ⅓×F for C, and 2⅓×F for E in eq.1: 3×F + 2×F + ⅓×F + 2×F + 2⅓×F + F = 32 which becomes 10⅔×F = 32 Divide both sides of the above equation by 10⅔: 10⅔×F ÷ 10⅔ = 32 ÷ 10⅔ which means F = 3 making A = 3×F = 3 × 3 = 9 B = D = 2×F = 2 × 3 = 6 C = ⅓×F = ⅓ × 3 = 1 E = 2⅓×F = 2⅓ × 3 = 7 and ABCDEF = 961673