Puzzle for June 8, 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.6, replace F with B + C (from eq.2): B + C – B = A – D which becomes C = A – D Add D to both sides of the above equation: C + D = A – D + D which becomes eq.6a) C + D = A
Hint #2
In eq.6a, replace A with C + E (from eq.3): C + D = C + E Subtract C from each side of the above equation: C + D – C = C + E – C which makes D = E
Hint #3
In eq.5, substitute D for E: D + D = F – D which becomes 2×D = F – D Add D to both sides of the equation above: 2×D + D = F – D + D which makes eq.5a) 3×D = F
Hint #4
Substitute 3×D for F (from eq.5a) in eq.2: 3×D = B + C Subtract B from each side of the equation above: 3×D – B = B + C – B which becomes eq.2a) 3×D – B = C
Hint #5
Substitute 3×D – B for C (from eq.2a) into eq.4: 3×D – B – B = B – D which becomes 3×D – 2×B = B – D Add 2×B and D to both sides of the above equation: 3×D – 2×B + 2×B + D = B – D + 2×B + D which makes 4×D = 3×B Divide both sides by 4: 4×D ÷ 4 = 3×B ÷ 4 which makes D = ¾×B and also makes E = D = ¾×B
Hint #6
Substitute (¾×B) for D in eq.5a: 3×(¾×B) = F which makes 2¼×B = F
Hint #7
Substitute (¾×B) for D in eq.2a: 3×(¾×B) – B = C which becomes 2¼×B – B = C which makes 1¼×B = C
Hint #8
Substitute 1¼×B for C, and ¾×B for E in eq.3: A = 1¼×B + ¾×B which makes A = 2×B
Solution
Substitute 2×B for A, 1¼×B for C, ¾×B for D for E, 2¼×B for F in eq.1: 2×B + B + 1¼×B + ¾×B + ¾×B + 2¼×B = 32 which simplifies to 8×B = 32 Divide both sides of the above equation by 8: 8×B ÷ 8 = 32 ÷ 8 which means B = 4 making A = 2×B = 2 × 4 = 8 C = 1¼×B = 1¼ × 4 = 5 D = E = ¾×B = ¾ × 4 = 3 F = 2¼×B = 2¼ × 4 = 9 and ABCDEF = 845339