Puzzle for May 18, 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.6, replace B + C with E + F (from eq.4): A – E + F = E + F + E which becomes A – E + F = 2×E + F In the above equation, add E to both sides, and subtract F from both sides: A – E + F + E – F = 2×E + F + E – F which makes A = 3×E
Hint #2
eq.6 may be written as: A – E + F = B + E + C In the equation above, replace A with 3×E, and B + E with D + F – B (from eq.5): 3×E – E + F = D + F – B + C which becomes 2×E + F = D + F – B + C Subtract F from both sides, and add B to both sides: 2×E + F – F + B = D + F – B + C – F + B which becomes eq.6a) 2×E + B = D + C
Hint #3
In eq.4, substitute C + D + E for F (from eq.2): E + C + D + E = B + C which becomes 2×E + C + D = B + C Subtract C from each side of the equation above: 2×E + C + D – C = B + C – C which becomes eq.4a) 2×E + D = B
Hint #4
Substitute 2×E + D for B (from eq.4a) in eq.6a: 2×E + 2×E + D = D + C which becomes 4×E + D = D + C Subtract D from each side of the above equation: 4×E + D – D = D + C – D which makes 4×E = C
Hint #5
Substitute 4×E for C, and 3×E for A in eq.3: 4×E – 3×E = B – 4×E which becomes E = B – 4×E Add 4×E to both sides of the above equation: E + 4×E = B – 4×E + 4×E which makes 5×E = B
Hint #6
Substitute 5×E for B in eq.4a: 2×E + D = 5×E Subtract 2×E from both sides of the equation above: 2×E + D – 2×E = 5×E – 2×E which makes D = 3×E
Hint #7
Substitute 5×E for B, and 3×E for D in eq.5: 5×E + E = 3×E + F – 5×E which becomes 6×E = –2×E + F Add 2×E to both sides of the above equation: 6×E + 2×E = –2×E + F + 2×E which becomes 8×E = F
Solution
Substitute 3×E for A and D, 5×E for B, 4×E for C, and 8×E for F in eq.1: 3×E + 5×E + 4×E + 3×E + E + 8×E = 24 which simplifies to 24×E = 24 Divide both sides by 24: 24×E ÷ 24 = 24 ÷ 24 which means E = 1 making A = D = 3×E = 3 × 1 = 3 B = 5×E = 5 × 1 = 5 C = 4×E = 4 × 1 = 4 F = 8×E = 8 × 1 = 8 and ABCDEF = 354318