Puzzle for March 2, 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
Add D to both sides of eq.4: F – D + D = E + D which becomes eq.4a) F = E + D In eq.3, replace F with E + D: E + E + D = B + D which becomes 2×E + D = B + D Subtract D from each side of the above equation: 2×E + D – D = B + D – D which makes 2×E = B
Hint #2
In eq.5, substitute A + D for B (from eq.2), and E + D for F (from eq.4a): A + D – E = A – (E + D) which is equivalent to A + D – E = A – E – D In the equation above, subtract A from each side, and add E to each side: A + D – E – A + E = A – E – D – A + E which simplifies to D = –D which makes D = 0
Hint #3
In eq.2, replace D with 0: B = A + 0 which makes B = A and also makes B = A = 2×E
Hint #4
In eq.4, replace D with 0: F – 0 = E which makes F = E
Hint #5
In eq.6, substitute 2×E for B and A: C – 2×E = 2×E ÷ E which becomes C – 2×E = 2 Add 2×E to both sides of the equation above: C – 2×E + 2×E = 2 + 2×E which makes eq.6a) C = 2 + 2×E
Solution
Substitute 2×E for A and B, 2 + 2×E for C (from eq.6a), 0 for D, and E for F in eq.1: 2×E + 2×E + 2 + 2×E + 0 + E + E = 26 which simplifies to 8×E + 2 = 26 Subtract 2 from each side of the above equation: 8×E + 2 – 2 = 26 – 2 which makes 8×E = 24 Divide both sides by 8: 8×E ÷ 8 = 24 ÷ 8 which means E = 3 making A = B = 2×E = 2×3 = 6 C = 2 + 2×E = 2 + 2×3 = 2 + 6 = 8 (from eq.6a) F = E = 3 and ABCDEF = 668033