Puzzle for April 18, 2020 ( )
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 E and F to both sides of eq.4: D – E + E + F = C – F + E + F which becomes eq.4a) D + F = C + E Add B and E to both sides of eq.5: C – B + B + E = A – E + B + E which becomes eq.5a) C + E = A + B
Hint #2
In eq.5a, replace C + E with D + F (from eq.4a): eq.5b) D + F = A + B
Hint #3
In eq.2, replace A + B with D + F (from eq.5b): E + F = D + F Subtract F from each side of the equation above: E + F – F = D + F – F which becomes D = E
Hint #4
In eq.4, substitute D for E: D – D = C – F which becomes 0 = C – F Add F to each side of the equation above: 0 + F = C – F + F which makes F = C
Hint #5
eq.1 may be re-written as: A + B + D + E + C + F = 36 Substitute C + F for A + B + D + E (from eq.6) in the equation above: C + F + C + F = 36 which becomes eq.1a) 2×C + 2×F = 36
Hint #6
Substitute C for F in eq.1a: 2×C + 2×C = 36 which becomes 4×C = 36 Divide both sides of the equation above by 4: 4×C ÷ 4 = 36 ÷ 4 which makes C = 9 and also makes F = C = 9
Hint #7
Substitute D for E, and 9 for F in eq.2: D + 9 = A + B Subtract A from both sides of the equation above: D + 9 – A = A + B – A which becomes eq.2a) D + 9 – A = B
Hint #8
Substitute D + 9 – A for B (from eq.2a) in eq.3: D + 9 – A – A = A – D which becomes D + 9 – 2×A = A – D Add 2×A and D to both sides of the equation above: D + 9 – 2×A + 2×A + D = A – D + 2×A + D which becomes 2×D + 9 = 3×A Divide both sides by 3: (2×D + 9) ÷ 3 = 3×A ÷ 3 which makes eq.3a) ⅔×D + 3 = A
Hint #9
Substitute 9 for C, ⅔×D + 3 for A (from eq.3a), and D for E in eq.5: 9 – B = ⅔×D + 3 – D which becomes 9 – B = 3 – ⅓×D In the above equation, add B and ⅓×D to both sides, and subtract 3 from both sides: 9 – B + B + ⅓×D – 3 = 3 – ⅓×D + B + ⅓×D – 3 which becomes 6 + ⅓×D = B which may be written as eq.5c) ⅓×D + 6 = B
Solution
Substitute ⅔×D + 3 for A (from eq.3a), ⅓×D + 6 for B (from eq.5c), 9 for C and F, and D for E in eq.1: ⅔×D + 3 + ⅓×D + 6 + 9 + D + D + 9 = 36 which simplifies to 3×D + 27 = 36 Subtract 27 from both sides of the equation above: 3×D + 27 – 27 = 36 – 27 which makes 3×D = 9 Divide both sides by 3: 3×D ÷ 3 = 9 ÷ 3 which means D = 3 making A = ⅔×D + 3 = (⅔×3) + 3 = 2 + 3 = 5 (from eq.3a) B = ⅓×D + 6 = (⅓×3) + 6 = 1 + 6 = 7 (from eq.5c) E = D = 3 and ABCDEF = 579339