Eight Knights

Eight dashing knights are in love with [the] kingdom’s princess. Starting on January 1, the first knight visits the princess every day. The second night visits the princess on January 2 and every second day thereafter. The third knight visits the princess on January 3 and every third day thereafter. The pattern continues for each of the 8 knights. What is the total number of knight visits up to and including the first day on which all 8 knights visit the princess?
Source: mathcontest.olemiss.edu 9/3/2012

SOLUTION

Let’s simplify the problem by assuming there are only 4 knights. We make a little spreadsheet to detect any pattern.

The first time all 4 knights visit the princess is on day 12 which is the least common multiple (LCM) of 1,2,3,4.
During those 12 days, the number of visits by each knight is as follows
Knight 1\!: 12 times \left ( 12\div 1=12\right )
Knight 2\!: 6 times \left ( 12\div 2=6\right )
Knight 3\!: 4 times \left ( 12\div 3=4\right )
Knight 4\!: 3 times \left ( 12\div 4=3\right )
Total number of visits
12+6+4+3=25

Applying the pattern to 8 knights
LCM of 1,2,3,4,5,6,7,8 = 840
During those 840 days, the number of visits by each knight is as follows
Knight 1\!: 840
Knight 2\!: 420
Knight 3\!: 280
Knight 4\!: 210
Knight 5\!: 168
Knight 6\!: 140
Knight 7\!: 120
Knight 8\!: 105
Total number of visits
840+420+280+210+168+140+120+105=2283

Answer: 2283.

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About mvtrinh

Retired high school math teacher.
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