Puzzle for March 18, 2019 ( )
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 C + F to both sides of eq.3: A - C + C + F = E - F + C + F which becomes A + F = E + C which may be written as A + F = C + E In eq.5, replace C + E with A + F: D + F = A + F Subtract F from both sides of the above equation: D + F - F = A + F - F which makes D = A
Hint #2
In eq.4, replace E with A + B (from eq.6): A + D = B + A + B which becomes A + D = 2×B + A Subtract A from each side of the above equation: A + D - A = 2×B + A - A which makes D = 2×B which also makes A = D = 2×B
Hint #3
In eq.4, substitute 2×B for A and D: 2×B + 2×B = B + E which becomes 4×B = B + E Subtract B from both sides of the above equation: 4×B - B = B + E - B which means 3×B = E
Hint #4
Substitute 2×B for D in eq.2: B + C = 2×B Subtract B from each side: B + C - B = 2×B - B which makes C = B
Hint #5
Substitute 2×B for D, B for C, and 3×B for E in eq.5: 2×B + F = B + 3×B which becomes 2×B + F = 4×B Subtract 2×B from both sides: 2×B + F - 2×B = 4×B - 2×B which becomes F = 2×B
Solution
Substitute 2×B for A and D and F, B for C, and 3×B for E in eq.1: 2×B + B + B + 2×B + 3×B + 2×B = 33 which simplifies to 11×B = 33 Divide both sides by 11: 11×B ÷ 11 = 33 ÷ 11 which means B = 3 making A = D = F = 2×B = 2 × 3 = 6 C = B = 3 E = 3×B = 3 × 3 = 9 and ABCDEF = 633696