Puzzle for April 20, 2023 ( )
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.
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Hint #1
In eq.5, substitute (A - F) for E (from eq.3): F - (A - F) = C - A which becomes F - A + F = C - A which becomes 2×F - A = C - A Add A to both sides of the above equation: 2×F - A + A = C - A + A which makes eq.5a) 2×F = C
Hint #2
Add B to both sides of eq.2: D + B = A - B + B which becomes eq.2a) D + B = A
Hint #3
eq.6 may be written as: E = (B + C + D + F) ÷ 4 Multiply both sides of the above equation by 4: 4 × E = 4 × (B + C + D + F) ÷ 4 which becomes 4×E = B + C + D + F which may be written as eq.6a) 4×E = D + B + C + F
Hint #4
In eq.6a, replace D + B with A (from eq.2a): eq.6b) 4×E = A + C + F
Hint #5
Add F to both sides of eq.3: E + F = A - F + F which becomes eq.3a) E + F = A
Hint #6
In eq.6b, substitute E + F for A (from eq.3a), and 2×F for C: 4×E = E + F + 2×F + F which becomes 4×E = E + 4×F Subtract E from each side of the equation above: 4×E - E = E + 4×F - E which becomes 3×E = 4×F Divide both sides by 4: 3×E ÷ 4 = 4×F ÷ 4 which makes ¾×E = F
Hint #7
Substitute (¾×E) for F in eq.5a: 2×(¾×E) = C which makes 1½×E = C
Hint #8
Substitute ¾×E for F in eq.3a: E + ¾×E = A which makes 1¾×E = A
Hint #9
Substitute 1¾×E for A in eq.4: B - E = E - 1¾×E which becomes B - E = -¾×E Add E to both sides of the above equation: B - E + E = -¾×E + E which makes B = ¼×E
Hint #10
Substitute 1¾×E for A, and ¼×E for B in eq.2: D = 1¾×E - ¼×E which makes D = 1½×E
Solution
Substitute 1¾×E for A, ¼×E for B, 1½×E for C and D, and ¾×E for F in eq.1: 1¾×E + ¼×E + 1½×E + 1½×E + E + ¾×E = 27 which simplifies to 6¾×E = 27 Divide both sides of the above equation by 6¾: 6¾×E ÷ 6¾ = 27 ÷ 6¾ which means E = 4 making A = 1¾×E = 1¾ × 4 = 7 B = ¼×E = ¼ × 4 = 1 C = D = 1½×E = 1½ × 4 = 6 F = ¾×E = ¾ × 4 = 3 and ABCDEF = 716643