Puzzle for April 5, 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 C + E with B + F (from eq.4): A + B = B + F + F which becomes A + B = B + 2×F Subtract B from each side of the equation above: A + B – B = B + 2×F – B which makes A = 2×F
Hint #2
In eq.5, replace A with 2×F, and B with D + F (from eq.2): C + D = 2×F + D + F + F which becomes C + D = 4×F + D Subtract D from each side of the above equation: C + D – D = 4×F + D – D which makes C = 4×F
Hint #3
In eq.3, substitute 4×F for C: E = 4×F + F which makes E = 5×F
Hint #4
Substitute 2×F for A, 4×F for C, and 5×F for E in eq.6: 2×F + B = 4×F + 5×F + F which becomes 2×F + B = 10×F Subtract 2×F from both sides of the above equation: 2×F + B – 2×F = 10×F – 2×F which makes B = 8×F
Hint #5
Substitute 8×F for B in eq.2: 8×F = D + F Subtract F from both sides of the equation above: 8×F – F = D + F – F which makes 7×F = D
Solution
Substitute 2×F for A, 8×F for B, 4×F for C, 7×F for D, and 5×F for E in eq.1: 2×F + 8×F + 4×F + 7×F + 5×F + F = 27 which simplifies to 27×F = 27 Divide both sides of the above equation by 27: 27×F ÷ 27 = 27 ÷ 27 which means F = 1 making A = 2×F = 2 × 1 = 2 B = 8×F = 8 × 1 = 8 C = 4×F = 4 × 1 = 4 D = 7×F = 7 × 1 = 7 E = 5×F = 5 × 1 = 5 and ABCDEF = 284751