Puzzle for December 22, 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
eq.2 may be re-written as: F – D = A + B + D + C In the above equation, replace A + B + D with F – E (from eq.3): F – D = F – E + C Add E and D to both sides: F – D + E + D = F – E + C + E + D which becomes F + E = F + C + D Subtract F from both sides: F + E – F = F + C + D – F which becomes eq.2a) E = C + D
Hint #2
In eq.5, replace E with C + D (from eq.2a): C + D = B + C Subtract C from each side of the equation above: C + D – C = B + C – C which makes D = B
Hint #3
In eq.4, add B to both sides, and subtract E from both sides: B + E + B – E = A – B + B – E which becomes 2×B = A – E Substitute B for D, and 2×B for A – E in eq.6: C + B = 2×B Subtract B from both sides of the equation above: C + B – B = 2×B – B which makes C = B
Hint #4
Substitute B for C and D in eq.2a: E = B + B which makes E = 2×B
Hint #5
Substitute 2×B for E in eq.4: B + 2×B = A – B which becomes 3×B = A – B Add B to each side of the equation above: 3×B + B = A – B + B which makes 4×B = A
Hint #6
Substitute 4×B for A, B for D, and 2×B for E in eq.3: 4×B + B + B = F – 2×B which becomes 6×B = F – 2×B Add 2×B to both sides of the equation above: 6×B + 2×B = F – 2×B + 2×B which makes 8×B = F
Solution
Substitute 4×B for A, B for C and D, 2×B for E, and 8×B for F in eq.1: 4×B + B + B + B + 2×B + 8×B = 17×B which simplifies to 17×B = 17 Divide both sides of the equation above by 17: 17×B ÷ 17 = 17 ÷ 17 which means B = 1 making A = 4×B = 4 × 1 = 4 C = D = B = 1 E = 2×B = 2 × 1 = 2 F = 8×B = 8 × 1 = 8 and ABCDEF = 411128