Puzzle for September 29, 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
In eq.4, replace A + E with D + F (from eq.3): B + D = D + F + F which becomes B + D = D + 2×F Subtract D from each side of the above equation: B + D – D = D + 2×F – D which makes B = 2×F
Hint #2
eq.6 may be written as: B = (A + D + E + F) ÷ 4 Multiply both sides of the above equation by 4: B × 4 = (A + D + E + F) ÷ 4 × 4 which becomes B × 4 = A + D + E + F which may be written as eq.6a) 4×B = A + E + D + F
Hint #3
In eq.6a, replace B with (2×F), and A + E with D + F (from eq.3): 4×(2×F) = D + F + D + F which becomes 8×F = 2×D + 2×F Subtract 2×F from both sides of the equation above: 8×F – 2×F = 2×D + 2×F – 2×F which becomes 6×F = 2×D Divide both sides by 2: 6×F ÷ 2 = 2×D ÷ 2 which makes 3×F = D
Hint #4
In eq.5, substitute 3×F for D, and 2×F for B: A + 3×F – E = 2×F + E + F which becomes A + 3×F – E = 3×F + E In the above equation, subtract 3×F from each side, and add E to each side: A + 3×F – E – 3×F + E = 3×F + E – 3×F + E which simplifies to eq.5a) A = 2×E
Hint #5
Substitute (2×F) for B, 2×E for A (from eq.5a), and 3×F for D in eq.6a: 4×(2×F) = 2×E + E + 3×F + F which becomes 8×F = 3×E + 4×F Subtract 4×F from both sides of the above equation: 8×F – 4×F = 3×E + 4×F – 4×F which becomes 4×F = 3×E Divide both sides by 3: 4×F ÷ 3 = 3×E ÷ 3 which makes 1⅓×F = E
Hint #6
Substitute (1⅓×F) for E in eq.5a: A = 2×(1⅓×F) which makes A = 2⅔×F
Hint #7
Substitute 1⅓×F for E, and 2×F for B in eq.2: C + 1⅓×F = 2×F Subtract 1⅓×F from each side of the equation above: C + 1⅓×F – 1⅓×F = 2×F – 1⅓×F which makes C = ⅔×F
Solution
Substitute 2⅔×F for A, 2×F for B, ⅔×F for C, 3×F for D, and 1⅓×F for E in eq.1: 2⅔×F + 2×F + ⅔×F + 3×F + 1⅓×F + F = 32 which simplifies to 10⅔×F = 32 Divide both sides of the above equation by 10⅔: 10⅔×F ÷ 10⅔ = 32 ÷ 10⅔ which means F = 3 making A = 2⅔×F = 2⅔ × 3 = 8 B = 2×F = 2 × 3 = 6 C = ⅔×F = ⅔ × 3 = 2 D = 3×F = 3 × 3 = 9 E = 1⅓×F = 1⅓ × 3 = 4 and ABCDEF = 862943