Puzzle for December 20, 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.
* AB is a 2-digit number (not A×B).
Scratchpad
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Hint #1
Add C and F to both sides of eq.3: B – C + C + F = E – F + C + F which becomes B + F = E + C which may be written as B + F = C + E In eq.2, replace C + E with B + F: B + B + F = A + D + F Subtract F from both sides of the equation above: B + B + F – F = A + D + F – F which becomes eq.2a) 2×B = A + D
Hint #2
Add D to both sides of eq.5: A + D = F – D + D which becomes A + D = F In eq.2a, replace A + D with F: eq.2b) 2×B = F
Hint #3
Subtract the left and right sides of eq.5 from the left and right sides of eq.4, respectively: D – A – A = E – (F – D) which is equivalent to D – 2×A = E – F + D In the equation above, subtract D from both sides, and add F to each side:: D – 2×A – D + F = E – F + D – D + F which becomes –2×A + F = E Substitute 2×B for F: –2×A + 2×B = E which may be written as eq.4a) 2×B – 2×A = E
Hint #4
eq.6 may be written as: E + F = 10×A + B Substitute 2×B – 2×A for E (from eq.4a), and 2×B for F: 2×B – 2×A + 2×B = 10×A + B which becomes 4×B – 2×A = 10×A + B In the above equation, add 2×A to both sides, and subtract B from each side: 4×B – 2×A + 2×A – B = 10×A + B + 2×A – B which simplifies to 3×B = 12×A Divide both sides by 3: 3×B ÷ 3 = 12×A ÷ 3 which makes B = 4×A
Hint #5
Substitute (4×A) for B in eq.4a: 2×(4×A) – 2×A = E which becomes 8×A – 2×A = E which makes 6×A = E
Hint #6
Substitute (4×A) for B in eq.2b: 2×(4×A) = F which makes 8×A = F
Hint #7
Substitute 8×A for F in eq.5: A = 8×A – D In the equation above, add D to each side, and subtract A from each side: A + D – A = 8×A – D + D – A which makes D = 7×A
Hint #8
Substitute 4×A for B, 6×A for E, and 8×A for F in eq.3: 4×A – C = 6×A – 8×A which becomes 4×A – C = –2×A Add C and 2×A to both sides of the equation above: 4×A – C + C + 2×A = –2×A + C + 2×A which simplifies to 6×A = C
Solution
Substitute 4×A for B, 6×A for C and E, 7×A for D, and 8×A for F in eq.1: A + 4×A + 6×A + 7×A + 6×A + 8×A = 32 which simplifies to 32×A = 32 Divide both sides of the equation above by 32: 32×A ÷ 32 = 32 ÷ 32 which means A = 1 making B = 4×A = 4×1 = 4 C = E = 6×A = 6×1 = 6 D = 7×A = 7×1 = 7 F = 8×A = 8×1 = 8 and ABCDEF = 146768