Puzzle for June 23, 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
In eq.5, replace A + C with B – C (from eq.4): B – C + D + F = B In the above equation, subtract B from both sides, and add C to both sides: B – C + D + F – B + C = B – B + C which simplifies to eq.5a) D + F = C
Hint #2
In eq.2, replace D + F with C (from eq.5a): C = A
Hint #3
In eq.4, substitute C for A: B – C = C + C which becomes B – C = 2×C Add C to both sides of the above equation: B – C + C = 2×C + C which makes B = 3×C
Hint #4
In eq.3, substitute C for A: C + D = C + E Subtract C from each side of the equation above: C + D – C = C + E – C which makes D = E
Hint #5
eq.6 may be written as: B – D = A + E + D In the above equation, substitute C + D for A + E (from eq.3): B – D = C + D + D whch becomes B – D = C + 2×D Add D to both sides: B – D + D = C + 2×D + D which becomes eq.6a) B = C + 3×D
Hint #6
Substitute 3×C for B in eq.6a: 3×C = C + 3×D Subtract C from each side of the above equation: 3×C – C = C + 3×D – C which makes 2×C = 3×D Divide both sides by 3: 2×C ÷ 3 = 3×D ÷ 3 which makes ⅔×C = D and also makes D = E = ⅔×C
Hint #7
Substitute ⅔×C for D in eq.5a: ⅔×C + F = C Subtract ⅔×C from each side of the equation above: ⅔×C + F – ⅔×C = C – ⅔×C which makes F = ⅓×C
Solution
Substitute C for A, 3×C for B, ⅔×C for D and E, and ⅓×C for F in eq.1: C + 3×C + C + ⅔×C + ⅔×C + ⅓×C = 20 which simplifies to 6⅔×C = 20 Divide both sides of the above equation by 6⅔: 6⅔×C ÷ 6⅔ = 20 ÷ 6⅔ which means C = 3 making A = C = 3 B = 3×C = 3 × 3 = 9 D = E = ⅔×C = ⅔ × 3 = 2 F = ⅓×C = ⅓ × 3 = 1 and ABCDEF = 393221