Puzzle for April 20, 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
In eq.3, replace E with C + D (from eq.4): C + D – F = B + C – D In the above equation, subtract C from both sides, and add D to both sides: C + D – F – C + D = B + C – D – C + D which becomes eq.3a) 2×D – F = B
Hint #2
Add C and D to both sides of eq.5: D – C + C + D = F – D + C + D which becomes 2×D = F + C In eq.3a, replace 2×D with F + C: F + C – F = B which makes C = B
Hint #3
In eq.6, substitute B for C: F – B = B + B – F which becomes F – B = 2×B – F Add B and F to both sides of the equation above: F – B + B + F = 2×B – F + B + F which makes 2×F = 3×B Divide both sides by 2: 2×F ÷ 2 = 3×B ÷ 2 which makes F = 1½×B
Hint #4
Substitute 1½×B for F in eq.3a: 2×D – 1½×B = B Add 1½×B to each side of the equation above: 2×D – 1½×B + 1½×B = B + 1½×B which becomes 2×D = 2½×B Divide both sides by 2: 2×D ÷ 2 = 2½×B ÷ 2 which makes D = 1¼×B
Hint #5
Substitute 1¼×B for D in eq.2: B + 1¼×B = A which makes 2¼×B = A
Hint #6
Substitute B for C, and 1¼×B for D in eq.4: B + 1¼×B = E which makes 2¼×B = E
Solution
Substitute 2¼×B for A and E, B for C, 1¼×B for D, and 1½×B for F in eq.1: 2¼×B + B + B + 1¼×B + 2¼×B + 1½×B = 37 which simplifies to 9¼×B = 37 Divide both sides of the equation above by 9¼: 9¼×B ÷ 9¼ = 37 ÷ 9¼ which means B = 4 making A = E = 2¼×B = 2¼ × 4 = 9 C = B = 4 D = 1¼×B = 1¼ × 4 = 5 F = 1½×B = 1½ × 4 = 6 and ABCDEF = 944596