Puzzle for July 4, 2023 ( )
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 B to both sides of eq.4: B + E + B = C - B + B which becomes eq.4a) 2×B + E = C
Hint #2
In eq.2, replace C with 2×B + E (from eq.4a): 2×B + E = E + F Subtract E from each side of the equation above: 2×B + E - E = E + F - E which makes 2×B = F
Hint #3
In eq.6, replace D with A + B + C + F (from eq.3), and F with 2×B: B + A + B + C + F = A + C + E + 2×B which becomes 2×B + A + C + F = A + C + E + 2×B Subtract 2×B, A, and C from each side of the above equation: 2×B + A + C + F - 2×B - A - C = A + C + E + 2×B - 2×B - A - C which makes F = E and also makes 2×B = F = E
Hint #4
Substitute 2×B for E in eq.4: B + 2×B = C - B which becomes 3×B = C - B Add B to both sides of the above equation: 3×B + B = C - B + B which makes 4×B = C
Hint #5
Substitute 4×B for C in eq.5: D - 4×B = B + 4×B which becomes D - 4×B = 5×B Add 4×B to both sides of the equation above: D - 4×B + 4×B = 5×B + 4×B which makes D = 9×B
Hint #6
Substitute 9×B for D, 4×B for C, and 2×B for F in eq.3: 9×B = A + B + 4×B + 2×B which becomes 9×B = A + 7×B Subtract 7×B from each side of the equation above: 9×B - 7×B = A + 7×B - 7×B which makes A = 2×B
Solution
Substitute 2×B for A and E and F, 4×B for C, and 9×B for D in eq.1: 2×B + B + 4×B + 9×B + 2×B + 2×B = 20 which simplifies to 20×B = 20 Divide both sides of the above equation by 20: 20×B ÷ 20 = 20 ÷ 20 which means B = 1 making A = E = F = 2×B = 2 × 1 = 2 C = 4×B = 4 × 1 = 4 D = 9×B = 9 × 1 = 9 and ABCDEF = 214922