Puzzle for June 24, 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
Add A and C to both sides of eq.6: F – A + A + C = B – C + D + A + C which becomes F + C = B + D + A which may be written as F + C = A + B + D In the above equation, replace A + B with C + D (from eq.3): F + C = C + D + D which becomes F + C = C + 2×D Subtract C from each side: F + C – C = C + 2×D – C which makes eq.6a) F = 2×D
Hint #2
In eq.5, replace E with B + D (from eq.2): B + D + F = A + B + C Subtract B from each side of the equation above: B + D + F – B = A + B + C – B which becomes eq.5a) D + F = A + C
Hint #3
In eq.4, substitute A + C for D + F (from eq.5a): A + C = B + C Subtract C from both sides of the above equation: A + C – C = B + C – C which makes A = B
Hint #4
Substitute 2×D for F in eq.5a: D + 2×D = A + C which becomes eq.5b) 3×D = A + C
Hint #5
Substitute 2×D for F, and A for B in eq.6: 2×D – A = A – C + D In the equation above, add A to both sides, and subtract D from both sides: 2×D – A + A – D = A – C + D + A – D which becomes eq.6b) D = 2×A – C
Hint #6
Substitute (2×A – C) for D (from eq.6b) in eq.5b: 3×(2×A – C) = A + C which becomes 6×A – 3×C = A + C In the equation above, add 3×C to both sides, and subtract A from both sides: 6×A – 3×C + 3×C – A = A + C + 3×C – A which makes 5×A = 4×C Divide both sides by 4: 5×A ÷ 4 = 4×C ÷ 4 which makes 1¼×A = C
Hint #7
Substitute 1¼×A for C in eq.6b: D = 2×A – 1¼×A which makes D = ¾×A
Hint #8
Substitute (¾×A) for D in eq.6a: F = 2×(¾×A) which makes F = 1½×A
Hint #9
Substitute A for B, and ¾×A for D in eq.2: A + ¾×A = E which makes 1¾×A = E
Solution
Substitute A for B, 1¼×A for C, ¾×A for D, 1¾×A for E, and 1½×A for F in eq.1: A + A + 1¼×A + ¾×A + 1¾×A + 1½×A = 29 which simplifies to 7¼×A = 29 Divide both sides of the equation above by 7¼: 7¼×A ÷ 7¼ = 29 ÷ 7¼ which means A = 4 making B = A = 4 C = 1¼×A = 1¼ × 4 = 5 D = ¾×A = ¾ × 4 = 3 E = 1¾×A = 1¾ × 4 = 7 F = 1½×A = 1½ × 4 = 6 and ABCDEF = 445376