Puzzle for October 1, 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.4, replace B with D + F (from eq.2): D + F + F = A + C which becomes eq.4a) D + 2×F = A + C
Hint #2
Add F and D to both sides of eq.3: E – F + F + D = C – D + F + D which becomes E + D = C + F In eq.5, replace C + F with E + D: E + D = A – C + E In the above equation, subtract E from both sides, and add C to both sides: E + D – E + C = A – C + E – E + C which becomes eq.5a) D + C = A
Hint #3
In eq.4a, substitute D + C for A (from eq.5a): D + 2×F = D + C + C which becomes D + 2×F = D + 2×C Subtract D from both sides of the above equation: D + 2×F – D = D + 2×C – D which becomes 2×F = 2×C Divide both sides by 2: 2×F ÷ 2 = 2×C ÷ 2 which makes F = C
Hint #4
Substitute C for F in eq.4: B + C = A + C Subtract C from each side of the equation above: B + C – C = A + C – C which makes B = A
Hint #5
Substitute C for F in eq.5: C + C = A – C + E which becomes 2×C = A – C + E In the equation above, subtract A from both sides, and add C to both sides: 2×C – A + C = A – C + E – A + C which becomes eq.5b) 3×C – A = E
Hint #6
Add the left and right sides of eq.3 to the left and right sides of eq.6, respectively: E + F – A + E – F = A + D + C – D which becomes 2×E – A = A + C Add A to both sides of the above equation: 2×E – A + A = A + C + A which becomes eq.6a) 2×E = 2×A + C
Hint #7
Substitute (3×C – A) for E (from eq.5b) into eq.6a: 2×(3×C – A) = 2×A + C which becomes 6×C – 2×A = 2×A + C In the equation above, add 2×A to both sides, and subtract C from both sides: 6×C – 2×A + 2×A – C = 2×A + C + 2×A – C which simplifies to 5×C = 4×A Divide both sides by 4: 5×C ÷ 4 = 4×A ÷ 4 which makes 1¼×C = A and also makes B = A = 1¼×C
Hint #8
Substitute 1¼×C for A in eq.5b: 3×C – 1¼×C = E which makes 1¾×C = E
Hint #9
Substitute 1¼×C for A in eq.5a: D + C = 1¼×C Subtract C from each side of the above equation: D + C – C = 1¼×C – C which makes D = ¼×C
Solution
Substitute 1¼×C for A and B, ¼×C for D, 1¾×C for E, and C for F in eq.1: 1¼×C + 1¼×C + C + ¼×C + 1¾×C + C = 26 which simplifies to 6½×C = 26 Divide both sides of the equation above by 6½: 6½×C ÷ 6½ = 26 ÷ 6½ which means C = 4 making A = B = 1¼×C = 1¼ × 4 = 5 D = ¼×C = ¼ × 4 = 1 E = 1¾×C = 1¾ × 4 = 7 F = C = 4 and ABCDEF = 554174