Puzzle for May 7, 2020 ( )
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.5, replace A with C + D (from eq.2): C + F = C + D + B Subtract C from both sides of the above equation: C + F – C = C + D + B – C which becomes eq.5a) F = D + B
Hint #2
In eq.3, replace F with D + B (from eq.5a): D + D + B = B which becomes 2×D + B = B Subtract B from each side of the equation above: 2×D + B – B = B – B which makes 2×D = 0 which means D = 0
Hint #3
In eq.5a, substitute 0 for D: F = 0 + B which makes F = B
Hint #4
Substitute 0 for D in eq.2: A = C + 0 which makes A = C
Hint #5
Substitute A for C, and B for F in eq.4: B – A + B = A + A – E which becomes 2×B – A = 2×A – E Add A and E to both sides of the equation above: 2×B – A + A + E = 2×A – E + A + E which becomes 2×B + E = 3×A Subtract 2×B from both sides: 2×B + E – 2×B = 3×A – 2×B which becomes eq.4a) E = 3×A – 2×B
Hint #6
Add E to both sides of eq.6: E – (D – E) + E = B – E + F + E which becomes E – D + E + E = B + F which becomes 3×E – D = B + F Substitute (3×A – 2×B) for E (from eq.4a), 0 for D, and B for F in the equation above: 3×(3×A – 2×B) – 0 = B + B which becomes 9×A – 6×B = 2×B Add 6×B to both sides: 9×A – 6×B + 6×B = 2×B + 6×B which becomes 9×A = 8×B Divide both sides by 8: 9×A ÷ 8 = 8×B ÷ 8 which makes 1⅛×A = B and also makes F = B = 1⅛×A
Hint #7
Substitute (1⅛×A) for B in eq.4a: E = 3×A – 2×(1⅛×A) which becomes E = 3×A – 2¼×A which makes E = ¾×A
Solution
Substitute 1⅛×A for B and F, A for C, 0 for D, and ¾×A for E in eq.1: A + 1⅛×A + A + 0 + ¾×A + 1⅛×A = 40 which simplifies to 5×A = 40 Divide both sides of the equation above by 5: 5×A ÷ 5 = 40 ÷ 5 which means A = 8 making B = F = 1⅛×A = 1⅛ × 8 = 9 C = A = 8 E = ¾×A = ¾ × 8 = 6 and ABCDEF = 898069