Puzzle for February 4, 2021 ( )
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
Add A to both sides of eq.6: F – A + A = A + C + D + A which becomes F = 2×A + C + D In eq.2, replace F with 2×A + C + D: B + C + D + E = 2×A + C + D Subtract C and D from each side of the above equation: B + C + D + E – C – D = 2×A + C + D – C – D which becomes eq.2a) B + E = 2×A
Hint #2
In eq.5, replace B + E with 2×A (from eq.2a): 2×A = A + C Subtract A from both sides of the above equation: 2×A – A = A + C – A which makes A = C
Hint #3
In eq.6, substitute A for C: F – A = A + A + D which becomes F – A = 2×A + D Add A to both sides of the above equation: F – A + A = 2×A + D + A which becomes eq.6a) F = 3×A + D
Hint #4
Substitute A for C, and 3×A + D for F (from eq.6a) in eq.4: A + A + D + E = 3×A + D – E which becomes 2×A + D + E = 3×A + D – E In the above equation, subtract 2×A and D from both sides, and add E to both sides: 2×A + D + E – 2×A – D + E = 3×A + D – E – 2×A – D + E which simplifies to 2×E = A and also makes C = A = 2×E
Hint #5
Substitute (2×E) for A in eq.2a: B + E = 2×(2×E) which becomes B + E = 4×E Subtract E from both sides of the equation above: B + E – E = 4×E – E which makes B = 3×E
Hint #6
Substitute 3×E for B, and 2×E for C in eq.3: F – 3×E = 3×E + 2×E which becomes F – 3×E = 5×E Add 3×E to both sides of the equation above: F – 3×E + 3×E = 5×E + 3×E which becomes F = 8×E
Hint #7
Substitute 8×E for F, and (2×E) for A in eq.6a: 8×E = 3×(2×E) + D which becomes 8×E = 6×E + D Subtract 6×E from both sides of the equation above: 8×E – 6×E = 6×E + D – 6×E which makes 2×E = D
Solution
Substitute 2×E for A and C and D, 3×E for B, and 8×E for F in eq.1: 2×E + 3×E + 2×E + 2×E + E + 8×E = 18 which simplifies to 18×E = 18 Divide both sides of the above equation by 18: 18×E ÷ 18 = 18 ÷ 18 which means E = 1 making A = C = D = 2×E = 2 × 1 = 2 B = 3×E = 3 × 1 = 3 F = 8×E = 8 × 1 = 8 and ABCDEF = 232218