Puzzle for May 4, 2022 ( )
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
Subtract A and D from both sides of eq.4: A + C – A – D = D + E – A – A – D which becomes eq.4a) C – D = E – 2×A Subtract A from both sides of eq.5: D + F – A = A + C – D – A which becomes eq.5a) D + F – A = C – D
Hint #2
In eq.4a, replace C – D with D + F – A (from eq.5a): D + F – A = E – 2×A Add 2×A to both sides of the above equation: D + F – A + 2×A = E – 2×A + 2×A which becomes eq.4b) D + F + A = E
Hint #3
In eq.3, replace E with D + F + A (from eq.4b): C + D + F + A = B – C + F In the equation above, subtract F from both sides, and add C to both sides: C + D + F + A – F + C = B – C + F – F + C which becomes 2×C + D + A = B which is the same as eq.3a) 2×C + A + D = B
Hint #4
In eq.3a, substitute B for A + D (from eq.2): 2×C + B = B Subtract B from each side of the above equation: 2×C + B – B = B – B which makes 2×C = 0 which means C = 0
Hint #5
Substitute A + D for B (from eq.2), and 0 for C in eq.6: A + F = A + D + 0 – F which becomes A + F = A + D – F In the equation above, subtract A from both sides, and add F to both sides: A + F – A + F = A + D – F – A + F which simplifies to 2×F = D
Hint #6
Substitute 2×F for D, and 0 for C in eq.5a: 2×F + F – A = 0 – 2×F which becomes 3×F – A = –2×F Add A and 2×F to both sides of the above equation: 3×F – A + A + 2×F = –2×F + A + 2×F which becomes 5×F = A
Hint #7
Substitute 5×F for A, and 2×F for D in eq.2: B = 5×F + 2×F which makes B = 7×F
Hint #8
Substitute 2×F for D, and 5×F for A in eq.4b: 2×F + F + 5×F = E which makes 8×F = E
Solution
Substitute 5×F for A, 7×F for B, 0 for C, 2×F for D, and 8×F for E in eq.1: 5×F + 7×F + 0 + 2×F + 8×F + F = 23 which simplifies to 23×F = 23 Divide both sides of the above equation by 23: 23×F ÷ 23 = 23 ÷ 23 which means F = 1 making A = 5×F = 5×1 = 5 B = 7×F = 7×1 = 7 D = 2×F = 2×1 = 2 E = 8×F = 8×1 = 8 and ABCDEF = 570281