Puzzle for July 24, 2021 ( )
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 B to both sides of eq.6: E + F – B + B = A + B + C + B which becomes eq.6a) E + F = A + 2×B + C Add F to both sides of eq.5: C + E – A + F = A + B – F + F which becomes C + E – A + F = A + B which may be written as eq.5a) C – A + E + F = A + B
Hint #2
In eq.5a, replace E + F with A + 2×B + C (from eq.6a): C – A + A + 2×B + C = A + B which becomes 2×B + 2×C = A + B Subtract B from both sides of the above equation: 2×B + 2×C – B = A + B – B which becomes eq.5b) B + 2×C = A
Hint #3
In eq.3, replace A with B + 2×C (from eq.5b): B + D = B + 2×C Subtract B from each side of the equation above: B + D – B = B + 2×C – B which makes D = 2×C
Hint #4
Add C to both sides of eq.4: A + C + C = E – C + C which becomes eq.4a) A + 2×C = E In eq.4a, substitute D for 2×C: eq.4b) A + D = E
Hint #5
Substitute F for A + D (from eq.2) in eq.4b: F = E
Hint #6
Substitute E for F, and B + 2×C for A (from eq.5b) in eq.6a: E + E = B + 2×C + 2×B + C which becomes eq.6b) 2×E = 3×B + 3×C
Hint #7
Substitute B + 2×C for A (from eq.5b) into eq.4a: B + 2×C + 2×C = E which becomes eq.4c) B + 4×C = E
Hint #8
Substitute (B + 4×C) for E (from eq.4c) into eq.6b: 2×(B + 4×C) = 3×B + 3×C which becomes 2×B + 8×C = 3×B + 3×C Subtract 2×B and 3×C from each side of the equation above: 2×B + 8×C – 2×B – 3×C = 3×B + 3×C – 2×B – 3×C which simplifies to 5×C = B
Hint #9
Substitute 5×C for B in eq.4c: 5×C + 4×C = E which makes 9×C = E and also makes F = E = 9×C
Hint #10
Substitute 5×C for B in eq.5b: 5×C + 2×C = A which makes 7×C = A
Solution
Substitute 7×C for A, 5×C for B, 2×C for D, and 9×C for E and F in eq.1: 7×C + 5×C + C + 2×C + 9×C + 9×C = 33 which simplifies to 33×C = 33 Divide both sides of the above equation by 33: 33×C ÷ 33 = 33 ÷ 33 which means C = 1 making A = 7×C = 7 × 1 = 7 B = 5×C = 5 × 1 = 5 D = 2×C = 2 × 1 = 2 E = F = 9×C = 9 × 1 = 9 and ABCDEF = 751299