Puzzle for April 7, 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 D + E with A – B (from eq.2): C + A – B = A + B In the equation above, subtract A from both sides, and add B to both sides: C + A – B – A + B = A + B – A + B which makes C = 2×B
Hint #2
In eq.5, replace C with 2×B: B + 2×B + D = A – B which becomes eq.5a) 3×B + D = A – B
Hint #3
In eq.5a, replace A – B with D + E (from eq.2): 3×B + D = D + E Subtract D from each side of the equation above: 3×B + D – D = D + E – D which makes 3×B = E
Hint #4
Add B to each side of eq.5a: 3×B + D + B = A – B + B which becomes eq.5b) 4×B + D = A Substitute 2×B for C, and 4×B + D for A (from eq.5b) in eq.6: F – 2×B + D = 4×B + D + B + 2×B which becomes F – 2×B + D = 7×B + D In the above equation, add 2×B to both sides, and subtract D from both sides: F – 2×B + D + 2×B – D = 7×B + D + 2×B – D which makes F = 9×B
Hint #5
Substitute 4×B + D for A (from eq.5a), 2×B for C, 9×B for F, and 3×B for E in eq.3: 4×B + D + 2×B = 9×B – 3×B which becomes 6×B + D = 6×B Subtract 6×B from each side of the above equation: 6×B + D – 6×B = 6×B – 6×B which makes D = 0
Hint #6
Substitute 0 for D in eq.5a: 4×B + 0 = A which makes 4×B = A
Solution
Substitute 4×B for A, 2×B for C, 0 for D, 3×B for E, and 9×B for F in eq.1: 4×B + B + 2×B + 0 + 3×B + 9×B = 19 which simplifies to 19×B = 19 Divide both sides of the equation above by 19: 19×B ÷ 19 = 19 ÷ 19 which means B = 1 making A = 4×B = 4×1 = 4 C = 2×B = 2×1 = 2 E = 3×B = 3×1 = 3 F = 9×B = 9×1 = 9 and ABCDEF = 412039