Puzzle for June 15, 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, substitute (C + F) for E (from eq.2): F – C – (C + F) = C – A which is equivalent to F – C – C – F = C – A which becomes –2×C = C – A Add 2×C and A to both sides of the above equation: –2×C + 2×C + A = C – A + 2×C + A which makes A = 3×C
Hint #2
In eq.3, replace 3×C for A, and E with C + F (from eq.2): B + F = 3×C + C + F which becomes B + F = 4×C + F Subtract F from both sides of the equation above B + F – F = 4×C + F – F which makes B = 4×C
Hint #3
Add F to both sides of eq.5: D + E – F + F = B + C + F which becomes D + E = B + C + F which may be written as D + E = B + F + C In the above equation, replace B + F with A + E (from eq.3): D + E = A + E + C Subtract E from each side of the equation above: D + E – E = A + E + C – E which becomes eq.5a) D = A + C
Hint #4
In eq.5a, substitute 3×C for A: D = 3×C + C which makes D = 4×C
Hint #5
Substitute 4×C for B and D in eq.6: C + E = 4×C + 4×C – E which becomes C + E = 8×C – E In the above equation, subtract C from each side, and add E to each side: C + E – C + E = 8×C – E – C + E which becomes 2×E = 7×C Divide both sides by 2: 2×E ÷ 2 = 7×C ÷ 2 which makes E = 3½×C
Hint #6
Substitute 3½×C for E in eq.2: 3½×C = C + F Subtract C from each side of the above equation: 3½×C – C = C + F – C which makes 2½×C = F
Solution
Substitute 3×C for A, 4×C for B and D, 3½×C for E, and 2½×C for F in eq.1: 3×C + 4×C + C + 4×C + 3½×C + 2½×C = 36 which simplifies to 18×C = 36 Divide both sides of the above equation by 18: 18×C ÷ 18 = 36 ÷ 18 which means C = 2 making A = 3×C = 3 × 2 = 6 B = D = 4×C = 4 × 2 = 8 E = 3½×C = 3½ × 2 = 7 F = 2½×C = 2½ × 2 = 5 and ABCDEF = 682875