Puzzle for June 24, 2022 ( )
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 A + B with D (from eq.2): eq.4a) C + E = D
Hint #2
In eq.5, replace D with C + E (from eq.4a): E + F = B + C + E Subtract E from each side of the equation above: E + F – E = B + C + E – E which becomes eq.5a) F = B + C
Hint #3
In eq.3, substitute B + C for F (from eq.5a): B + B + C = A + E which becomes 2×B + C = A + E Subtract 2×B and E from each side of the above equation: 2×B + C – 2×B – E = A + E – 2×B – E which becomes eq.3a) C – E = A – 2×B
Hint #4
Subtract the left and right sides of eq.3a from the left and right sides of eq.4, respectively: C + E – (C – E) = A + B – (A – 2×B) which becomes C + E – C + E = A + B – A + 2×B which makes 2×E = 3×B Divide both sides of the above equation by 2: 2×E ÷ 2 = 3×B ÷ 2 which makes E = 1½×B
Hint #5
Substitute 1½×B for E in eq.5: 1½×B + F = B + D Subtract B from each side of the above equation: 1½×B + F – B = B + D – B which makes eq.5b) ½×B + F = D
Hint #6
Substitute 1½×B for E, and ½×B + F for D (from eq.5b) in eq.6: B + 1½×B = ½×B + F – 1½×B + F which becomes 2½×B = –B + 2×F Add B to both sides of the equation above: 2½×B + B = –B + 2×F + B which makes 3½×B = 2×F Divide both sides by 2: 3½×B ÷ 2 = 2×F ÷ 2 which makes 1¾×B = F
Hint #7
Substitute 1¾×B for F in eq.5b: ½×B + 1¾×B = D which makes 2¼×B = D
Hint #8
Substitute 2¼×B for D in eq.2: 2¼×B = A + B Subtract B from both sides of the above equation: 2¼×B – B = A + B – B which makes 1¼×B = A
Hint #9
Substitute 1¾×B for F in eq.5a: 1¾×B = B + C Subtract B from each side of the above equation: 1¾×B – B = B + C – B which makes ¾×B = C
Solution
Substitute 1¼×B for A, ¾×B for C, 2¼×B for D, 1½×B for E, and 1¾×B for F in eq.1: 1¼×B + B + ¾×B + 2¼×B + 1½×B + 1¾×B = 34 which simplifies to 8½×B = 34 Divide both sides of the above equation by 8½: 8½×B ÷ 8½ = 34 ÷ 8½ which means B = 4 making A = 1¼×B = 1¼ × 4 = 5 C = ¾×B = ¾ × 4 = 3 D = 2¼×B = 2¼ × 4 = 9 E = 1½×B = 1½ × 4 = 6 F = 1¾×B = 1¾ × 4 = 7 and ABCDEF = 543967