Puzzle for May 27, 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.4, replace A with D + E (from eq.2), and replace D with C – F (from eq.3): D + E – B + C = C – F + E + F which becomes D + E – B + C = C + E In the above equation, subtract E and C from both sides, and add B to both sides: D + E – B + C – E – C + B = C + E – E – C + B which simplifies to D = B
Hint #2
In eq.5, replace D with B: E – C – B – F = B + C + B which becomes E – C – B – F = 2×B + C Add C, B, and F to both sides of the above equation: E – C – B – F + C + B + F = 2×B + C + C + B + F which becomes eq.5a) E = 3×B + 2×C + F
Hint #3
In eq.6, replace D with B: B + C + B + F = A – B – C – F which becomes 2×B + C + F = A – B – C – F Add B, C, and F to both sides of the equation above: 2×B + C + F + B + C + F = A – B – C – F + B + C + F which becomes 3×B + 2×C + 2×F = A which may be written as eq.6a) A = 3×B + 2×C + 2×F
Hint #4
Subtract the left and right sides of eq.5a from the left and right sides of eq.6a, respectively: A – E = 3×B + 2×C + 2×F – (3×B + 2×C + F) which becomes A – E = 3×B + 2×C + 2×F – 3×B – 2×C – F which becomes A – E = F Add E to both sides of the equation above: A – E + E = F + E which becomes eq.6b) A = F + E
Hint #5
Substitute F + E for A (from eq.6b) into eq.2: F + E = D + E Subtract E from each side of the above equation: F + E – E = D + E – E which makes F = D
Hint #6
Substitute D for F in eq.3: D = C – D Add D to both sides of the above equation: D + D = C – D + D which becomes 2×D = C
Hint #7
Substitute D for B and F, and (2×D) for C in eq.5a: E = 3×D + 2×(2×D) + D which becomes E = 3×D + 4×D + D which becomes E = 8×D
Hint #8
Substitute 8×D for E in eq.2: A = D + 8×D which makes A = 9×D
Solution
Substitute 9×D for A, D for B and F, 2×D for C, and 8×D for E in eq.1: 9×D + D + 2×D + D + 8×D + D = 22 which simplifies to 22×D = 22 Divide both sides of the above equation by 22: 22×D ÷ 22 = 22 ÷ 22 which means D = 1 making A = 9×D = 9 × 1 = 9 B = F = D = 1 C = 2×D = 2 × 1 = 2 E = 8×D = 8 × 1 = 8 and ABCDEF = 912181