Puzzle for July 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
Add C to both sides of eq.5: D + F – C + C = B + E + C which becomes D + F = B + E + C which is the same as eq.5a) D + F = B + C + E In eq.5a, replace C + E with A + F – E (from eq.4): D + F = B + A + F – E In the above equation, subtract F from both sides, and add E to both sides: D + F – F + E = B + A + F – E – F + E which becomes D + E = B + A which may be written as eq.4a) D + E = A + B
Hint #2
In eq.2, replace A + B with D + E (from eq.4a): C + D = D + E Subtract D from each side of the above equation: C + D – D = D + E – D which makes C = E
Hint #3
In eq.4, substitute C for E: A + F – C = C + C which becomes A + F – C = 2×C In the above equation, subtract A from both sides, and add C to both sides: A + F – C – A + C = 2×C – A + C which becomes eq.4b) F = 3×C – A
Hint #4
eq.6 may be written as: A = (C + E + F) ÷ 3 Multiply both sides of the above equation by 3: 3 × A = 3 × ((C + E + F) ÷ 3) which becomes 3×A = C + E + F Replace E with C: 3×A = C + C + F which becomes eq.6a) 3×A = 2×C + F
Hint #5
Substitute 3×C – A for F (from eq.4b) in eq.6a: 3×A = 2×C + 3×C – A which becomes 3×A = 5×C – A Add A to both sides of the equation above: 3×A + A = 5×C – A + A which makes 4×A = 5×C Divide both sides by 4: 4×A ÷ 4 = 5×C ÷ 4 which makes A = 1¼×C
Hint #6
Substitute 1¼×C for A in eq.4b: F = 3×C – 1¼×C which makes F = 1¾×C
Hint #7
Substitute 1¼×C for A in eq.2: 1¼×C + B = C + D Subtract C from each side of the equation above: 1¼×C + B – C = C + D – C which becomes eq.2a) ¼×C + B = D
Hint #8
Substitute 1¼×C for A, and 1¾×C for F in eq.3: B + D – 1¼×C = 1¼×C + 1¾×C which becomes B + D – 1¼×C = 3×C Add 1¼×C to both sides of the equation above: B + D – 1¼×C + 1¼×C = 3×C + 1¼×C which makes eq.3a) B + D = 4¼×C
Hint #9
Substitute ¼×C + B for D (from eq.2a) in eq.3a: B + ¼×C + B = 4¼×C which becomes 2×B + ¼×C = 4¼×C Subtract ¼×C from both sides of the above equation: 2×B + ¼×C – ¼×C = 4¼×C – ¼×C which makes 2×B = 4×C Divide both sides by 2: 2×B ÷ 2 = 4×C ÷ 2 which makes B = 2×C
Hint #10
Substitute 2×C for B in eq.2a: ¼×C + 2×C = D which makes 2¼×C = D
Solution
Substitute 1¼×C for A, 2×C for B, 2¼×C for D, C for E, and 1¾×C for F in eq.1: 1¼×C + 2×C + C + 2¼×C + C + 1¾×C = 37 which simplifies to 9¼×C = 37 Divide both sides of the above equation by 9¼: 9¼×C ÷ 9¼ = 37 ÷ 9¼ which means C = 4 making A = 1¼×C = 1¼ × 4 = 5 B = 2×C = 2 × 4 = 8 D = 2¼×C = 2¼ × 4 = 9 E = C = 4 F = 1¾×C = 1¾ × 4 = 7 and ABCDEF = 584947