Puzzle for July 29, 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
Subtract B and E from both sides of eq.4: D + E – B – E = A + B – B – E which becomes eq.4a) D – B = A – E Subtract E from both sides of eq.5: B + E – E = A – B + D – E which becomes B = A – B + D – E which may be written as eq.5a) B = A – E – B + D
Hint #2
In eq.5a, replace A – E with D – B (from eq.4a): B = D – B – B + D which becomes B = 2×D – 2×B Add 2×B to both sides of the above equation: B + 2×B = 2×D – 2×B + 2×B which becomes 3×B = 2×D Divide both sides by 2: 3×B ÷ 2 = 2×D ÷ 2 which makes 1½×B = D
Hint #3
Subtract the left and right sides of eq.4a from the left and right sides of eq.6, respectively: C + D – (D – B) = A – C + E – (A – E) which is equivalent to C + D – D + B = A – C + E – A + E which becomes C + B = –C + 2×E Add C to both sides of the above equation: C + B + C = –C + 2×E + C which becomes eq.6a) B + 2×C = 2×E
Hint #4
In eq.2, substitute 1½×B for D: 1½×B + F = B + E Subtract B from each side of the equation above: 1½×B + F – B = B + E – B which becomes eq.2a) ½×B + F = E
Hint #5
Substitute (½×B + F) for E (from eq.2a) into eq.6a: B + 2×C = 2(½×B + F) which becomes B + 2×C = B + 2×F Subtract B from each side of the equation above: B + 2×C – B = B + 2×F – B which becomes 2×C = 2×F Divide both sides by 2: 2×C ÷ 2 = 2×F ÷ 2 which makes C = F
Hint #6
Substitute C for F, and 1½×B for D in eq.3: C + C = B + 1½×B which becomes 2×C = 2½×B Divide both sides of the above equation by 2: 2×C ÷ 2 = 2½×B ÷ 2 which makes C = 1¼×B and also makes F = C = 1¼×B
Hint #7
Substitute 1¼×B for F in eq.2a: ½×B + 1¼×B = E which makes 1¾×B = E
Hint #8
Substitute 1½×B for D, and 1¾×B for E in eq.4: 1½×B + 1¾×B = A + B which becomes 3¼×B = A + B Subtract B from both sides of the equation above: 3¼×B – B = A + B – B which makes 2¼×B = A
Solution
Substitute 2¼×B for A, 1¼×B for C and F, 1½×B for D, and 1¾×B for E in eq.1: 2¼×B + B + 1¼×B + 1½×B + 1¾×B + 1¼×B = 36 which simplifies to 9×B = 36 Divide both sides of the above equation by 9: 9×B ÷ 9 = 36 ÷ 9 which means B = 4 making A = 2¼×B = 2¼ × 4 = 9 C = F = 1¼×B = 1¼ × 4 = 5 D = 1½×B = 1½ × 4 = 6 E = 1¾×B = 1¾ × 4 = 7 and ABCDEF = 945675