Puzzle for August 19, 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
Help Area
Hint #1
In eq.5, replace D + F with A (from eq.2): A + C = B + A Subtract A from each side of the above equation: A + C – A = B + A – A which makes C = B
Hint #2
Add A to both sides of eq.3: C + F + A = A + E + A which is the same as C + A + F = 2×A + E In the above equation, replace A + F with B + C + E (from eq.4): C + B + C + E = 2×A + E which becomes eq.3a) B + 2×C + E = 2×A + E
Hint #3
In eq.3a, substitute B for C: B + 2×B + E = 2×A + E which becomes 3×B + E = 2×A + E Subtract E from each side of the above equation: 3×B + E – E = 2×A + E – E which becomes 3×B = 2×A Divide both sides by 2: 3×B ÷ 2 = 2×A ÷ 2 which makes 1½×B = A
Hint #4
eq.6 may be written as: F = (A + B + C) ÷ 3 Multiply both sides of the above equation by 3: 3 × F = 3 × ((A + B + C) ÷ 3) which becomes eq.6a) 3×F = A + B + C
Hint #5
Substitute 1½×B for A, and B for C in eq.6a: 3×F = 1½×B + B + B which becomes 3×F = 3½×B Divide both sides of the above equation by 3: 3×F ÷ 3 = 3½×B ÷ 3 which makes F = 1⅙×B
Hint #6
Substitute 1⅙×B for F, and 1½×B for A in eq.2: D + 1⅙×B = 1½×B Subtract 1⅙×B from each side of the equation above: D + 1⅙×B – 1⅙×B = 1½×B – 1⅙×B which makes D = ⅓×B
Hint #7
Substitute B for C, 1⅙×B for F, and 1½×B for A in eq.3: B + 1⅙×B = 1½×B + E which becomes 2⅙×B = 1½×B + E Subtract 1½×B from each side of the equation above: 2⅙×B – 1½×B = 1½×B + E – 1½×B which makes ⅔×B = E
Solution
Substitute 1½×B for A, B for C, ⅓×B for D, ⅔×B for E, and 1⅙×B for F in eq.1: 1½×B + B + B + ⅓×B + ⅔×B + 1⅙×B = 34 which simplifies to 5⅔×B = 34 Divide both sides of the above equation by 5⅔: 5⅔×B ÷ 5⅔ = 34 ÷ 5⅔ which means B = 6 making A = 1½×B = 1½ × 6 = 9 C = B = 6 D = ⅓×B = ⅓ × 6 = 2 E = ⅔×B = ⅔ × 6 = 4 F = 1⅙×B = 1⅙ × 6 = 7 and ABCDEF = 966247