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### Not Dozens

Chris adds up all of the positive whole numbers less than 1,728 that are not integer multiples of twelve.

*What sum does Chris obtain?*

### Festive Factorial

To ring in the New Year, here’s a problem about the number 2017.

*What is the last non-zero digit of 2017! (the factorial of 2017 = 2017 × 2016 × … × 2 × 1)?*

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## Solutions to week 68

**Snug Circle**. As you can see in the diagram, there are two possible positions for circle *a*, centered on either point *A* or point *A*‘. However, the two positions are symmetric with respect to the *y*-axis, so the coordinates of point *A* will have the desired absolute values. The *y*-coordinate is easy; since radius *AC* is perpendicular to the *x*-axis and has length two, the *y*-coordinate of *A* is two. To get the *x*-coordinate, we need the length of horizontal line *AD*. But that is one leg of a right triangle with other leg *BD* of length 6 – 2 = 4, and hypotenuse *BA* of length 6 + 2 = 8. Therefore, the length of *AD* is √(64 - 16) = √48 = 4√3, and the desired coordinates are **(4√3, 2)**.

**Congruence Time**. The minute hand on a clock makes a complete circuit of 360° in 60 minutes, so it travels at 6° per minute, and since both minute hands start pointing straight up vertically, each minute hand points 6*m* degrees clockwise of vertical at *m* minutes after midnight in Los Angeles. The hour hand makes a complete circuit in 12 hours, which is 720 minutes, and so it moves at half a degree per minute. Thus, the Los Angeles hour hand points *m/2* degrees clockwise of vertical at *m* minutes after midnight. Since the New York clock starts with the hour hand pointing to the three, which is 90° clockwise of vertical, at that same time the New York hour hand points 90 + *m*/2 degrees clockwise of vertical. When they form the same angle, we must have 6*m* – *m*/2 = 90 + *m*/2 – 6*m*. Solving, this is 11*m* = 90, which means that the two clocks make the same angle **8 2/11** minutes after midnight in Los Angeles.

## Recent Weeks

**Week 68**: Snug Circle & Congruence Time, solutions to Card 61 & Card T

**Week 67**: Card 61 & Card T, solutions to Long Haul & Postal Pathways

**Week 66**: Long Haul & Postal Pathways, solutions to Card 129 & Card 94

**Week 65**: Card 129 & Card 94, solutions to Skimpy Schedule & Passing Pennies

**Week 64**: Skimpy Schedule & Passing Pennies, solutions to Seating Arrangement & Stuck on Fairness

Links to all of the puzzles and solutions are on the Complete Varsity Math page.

**Come back next week** for answers and more puzzles.