Puzzle for September 17, 2019 ( )
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.2, replace F with C – D + E (from eq.3): D + E = C + C – D + E In the above equation, subtract E from both sides, and add D to both sides: D + E – E + D = C + C – D + E – E + D which simplifies to 2×D = 2×C Divide both sides by 2: 2×D ÷ 2 = 2×C ÷ 2 which makes D = C
Hint #2
In eq.3, replace D with C: F = C – C + E which makes F = E
Hint #3
Add both A and B to each side of eq.5: E – A – B + A + B = A + B + A + B which becomes eq.5a) E = 2×A + 2×B
Hint #4
In eq.6, substitute 2×A + 2×B for E (from eq.5a): B + D = 2×A + 2×B – B which becomes B + D = 2×A + B Subtract B from both sides of the equation above: B + D – B = 2×A + B – B which makes D = 2×A which also makes eq.6a) C = D = 2×A
Hint #5
Substitute 2×A for both C and D, and 2×A + 2×B for E (from eq.5a) in eq.4: 2×A + 2×A = A + B + 2×A + 2×B which becomes 4×A = 3×A + 3×B Subtract 3×A from each side of the above equation: 4×A – 3×A = 3×A + 3×B – 3×A which makes A = 3×B
Hint #6
Substitute (3×B) for A in eq.5a: E = 2×(3×B) + 2×B which becomes E = 6×B + 2×B which means E = 8×B and also means F = E = 8×B
Hint #7
Substitute (3×B) for A in eq.6a: C = D = 2×(3×B) which makes C = D = 6×B
Solution
Substitute 3×B for A, 6×B for C and D, and 8×B for E and F in eq.1: 3×B + B + 6×B + 6×B + 8×B + 8×B = 32 which simplifies to 32×B = 32 Divide both sides of the equation above by 32: 32×B ÷ 32 = 32 ÷ 32 which means B = 1 making A = 3×B = 3 × 1 = 1 C = D = 6×B = 6 × 1 = 6 E = F = 8×B = 8 × 1 = 8 and ABCDEF = 316688