Puzzle for June 16, 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.6, replace D + E with A (from eq.2): A + C + A = F which becomes eq.6a) 2×A + C = F Add C to both sides of eq.5: F – C + C = A + C + C which becomes eq.5a) F = A + 2×C
Hint #2
In eq.6a, replace F with A + 2×C (from eq.5a): 2×A + C = A + 2×C Subtract A and C from both sides of the above equation: 2×A + C – A – C = A + 2×C – A – C which makes A = C
Hint #3
In eq.5a, substitute A for C: F = A + 2×A which becomes F = 3×A
Hint #4
Substitute 3×A for F into eq.3: eq.3a) B + D = 3×A
Hint #5
Substitute A for C into eq.4: A + D = B – D Add D to both sides of the equation above: A + D + D = B – D + D which becomes eq.4a) A + 2×D = B
Hint #6
Substitute A + 2×D for B (from eq.4a) in eq.3a: A + 2×D + D = 3×A which becomes A + 3×D = 3×A Subtract A from each side of the equation above: A + 3×D – A = 3×A – A which becomes 3×D = 2×A Divide both sides by 3: 3×D ÷ 3 = 2×A ÷ 3 which makes D = ⅔×A
Hint #7
Substitute ⅔×A for D in eq.3a: B + ⅔×A = 3×A Subtract ⅔×A from both sides of the above equation: B + ⅔×A – ⅔×A = 3×A – ⅔×A which makes B = 2⅓×A
Hint #8
Substitute ⅔×A for D in eq.2: ⅔×A + E = A Subtract ⅔×A from each side of the above equation: ⅔×A + E – ⅔×A = A – ⅔×A which makes E = ⅓×A
Solution
Substitute 2⅓×A for B, A for C, ⅔×A for D, ⅓×A for E, and 3×A for F in eq.1: A + 2⅓×A + A + ⅔×A + ⅓×A + 3×A = 25 which simplifies to 8⅓×A = 25 Divide both sides of the equation above by 8⅓: 8⅓×A ÷ 8⅓ = 25 ÷ 8⅓ which means A = 3 making B = 2⅓×A = 2⅓ × 3 = 7 C = A = 3 D = ⅔×A = ⅔ × 3 = 2 E = ⅓×A = ⅓ × 3 = 1 F = 3×A = 3 × 3 = 9 and ABCDEF = 373219