Puzzle for October 15, 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 D to both sides of eq.4: E + F + D = A + B – D + D which becomes E + F + D = A + B In eq.5, replace A + B with E + F + D: E + F + D + D – E = C + E which becomes eq.5a) F + 2×D = C + E
Hint #2
In eq.3, add D to both sides, and subtract E from both sides: E – F + D – E = C – D + D – E which becomes –F + D = C – E which may be written as eq.3a) D – F = C – E
Hint #3
Add the left and right sides of eq.3a to the left and right sides of eq.5a, respectively: F + 2×D + D – F = C + E + C – E which becomes 3×D = 2×C Divide both sides of the equation above by 2: 3×D ÷ 2 = 2×C ÷ 2 which makes 1½×D = C
Hint #4
In eq.6, replace C with 1½×D: E ÷ F = 1½×D ÷ D which becomes E ÷ F = 1½ Multiply both sides of the above equation by F: E ÷ F × F = 1½ × F which makes eq.6a) E = 1½×F
Hint #5
In eq.3, substitute 1½×F for E, and 1½×D for C: 1½×F – F = 1½×D – D which makes ½×F = ½×D Multiply both sides of the equation above by 2: 2 × ½×F = 2 × ½×D which makes F = D
Hint #6
Substitute D for F in eq.6a: E = 1½×D which also means E = C = 1½×D
Hint #7
Substitute E for C in eq.2: B – E = A – E Add E to both sides of the above equation: B – E + E = A – E + E which makes B = A
Hint #8
Substitute 1½×D for E, D for F, and A for B in eq.4: 1½×D + D = A + A – D which becomes 2½×D = 2×A – D Add D to both sides of the equation above: 2½×D + D = 2×A – D + D which makes 3½×D = 2×A Divide both sides by 2: 3½×D ÷ 2 = 2×A ÷ 2 which makes 1¾×D = A and also makes B = A = 1¾×D
Solution
Substitute 1¾×D for A and B, 1½×D for C and E, and D for F in eq.1: 1¾×D + 1¾×D + 1½×D + D + 1½×D + D = 34 which simplifies to 8½×D = 34 Divide both sides of the equation above by 8½: 8½×D ÷ 8½ = 34 ÷ 8½ which means D = 4 making A = B = 1¾×D = 1¾ × 4 = 7 C = E = 1½×D = 1½ × 4 = 6 F = D = 4 and ABCDEF = 776464