Puzzle for December 7, 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 and EF are 2-digit numbers (not A×B or E×F).
Scratchpad
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Hint #1
Add B to both sides of eq.2: E + B = A – B + B which becomes E + B = A In eq.4, replace A with (E + B): C – E = F – (E + B) which is equivalent to C – E = F – E – B Add E and B to both sides of the equation above: C – E + E + B = F – E – B + E + B which becomes C + B = F which is the same as eq.4a) B + C = F
Hint #2
In eq.1, substitute F for both B + C (from eq.4a) and for D + E (from eq.3): A + F + F + F = 29 which becomes eq.1a) A + 3×F = 29
Hint #3
eq.5 may be written as: 10×E + F + A = 10×A + B In the equation above, substitute (A – B) for E (from eq.2): 10×(A – B) + F + A = 10×A + B which is the same as 10×A – 10×B + F + A = 10×A + B Subtract 10×A from both sides, and add 10×B to each side: 10×A – 10×B + F + A – 10×A + 10×B = 10×A + B – 10×A + 10×B which becomes F + A = 11×B which may be written as eq.5a) A + F = 11×B
Hint #4
To make eq.5a true, check several possible values for B, A, and F: If B = 0, then A + F = 11×0 = 0 If B = 1, then A + F = 11×1 = 11 If B = 2, then A + F = 11×2 = 22 If B > 2, then A + F > 22 Since A and F must be one-digit non-negative integers, then A + F ≤ 18 Therefore: B = 1 and A + F = 11 or: B = 0 and A + F = 0 which would make A = F = 0
Hint #5
Check: B = 0, and A = F = 0 ... Substituting 0 for A and F in eq.1a would yield: 0 + 3×0 = 29 which would mean 0 = 29 Since 0 ≠ 29, then B ≠ 0 and A = F ≠ 0 which means B = 1 and eq.5b) A + F = 11
Hint #6
eq.1a may be written as: A + F + 2×F = 29 Substitute 11 for A + F (from eq.5b) in the equation above: 11 + 2×F = 29 Subtract 11 from both sides: 11 + 2×F – 11 = 29 – 11 which makes 2×F = 18 Divide both sides by 2: 2×F ÷ 2 = 18 ÷ 2 which makes F = 9
Hint #7
Substitute 9 for F in eq.5b: A + 9 = 11 Subtract 9 from both sides of the above equation: A + 9 – 9 = 11 – 9 which makes A = 2
Hint #8
Substitute 2 for A, and 1 for B in eq.2: E = 2 – 1 which makes E = 1
Hint #9
Substitute 1 for E, 9 for F, and 2 for A in eq.4: C – 1 = 9 – 2 which means C – 1 = 7 Add 1 to each side of the equation above: C – 1 + 1 = 7 + 1 which makes C = 8
Solution
Substitute 9 for F, and 1 for E in eq.3: 9 = D + 1 Subtract 1 from each side of the above equation: 9 – 1 = D + 1 – 1 which makes 8 = D and ABCDEF = 218819