## ________________

Points and lines are the theme this week.

## ________________

### Maximizing Links

Define a link as a straight line connecting any two points of the six points in the figure

to the right.

*How many links can be placed without forming a triangle having three of the pointsas vertices?*

### Seven Points

*Place seven points at different positions on the same plane so that for any three chosen, at least two will be exactly one foot apart.*

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## Solutions to week 132

The time in Time of Day is **8:24 am**. The simple continuous function in The Missing Element is **F(x) = 120(2 ^{x} – 1)/x**. The terms shown are for

**x = -3, -2, -1, 0, 1, 2, 3, 4, 5, 6. q = F(0) = 120 × ln2 which is approximately equal to 83.17766**.

**Time of Day answer explained:**

For the hour hand to be exactly on a minute, the second hand must be on the 12. If the second hand is 18 minute marks ahead of the hour hand, the hour hand is on the 42nd minute mark and the time is 8:24. It must be 8:24 am because it would not be daylight at 8:24 pm.

**The Missing Element answer explained:**

The sequence can be expressed by the simple continuous function, F(x) = 120(2^{x} – 1)/x when x is not equal to 0. The terms shown are for x = -3, -2, -1, 0, 1, 2, 3, 4, 5, 6. To get q, take the limit of F(x) as x approaches 0. When x is very near 0, 2^{x} = e^{xln2} ≈ 1 + xln2 where ln2 is the natural logarithm of 2 ≈ .6931472. This produces q = 120 × ln2 ≈ 83.17766.

## Recent Weeks

**Week 132**: Time of Day & The Missing Element, solutions to Knight Trap & My Number is 136

**Week 131**: Knight Trap & My Number is 136, solutions to The Size of Humanity & Fish Pond

**Week 130**: The Size of Humanity & Fish Pond, solutions to Circumnavigation & Typos

**Week 129**: Circumnavigation & Typos, solutions to Keeping Things Even & Roll Those Bones

**Week 128**: Keeping Things Even & Roll Those Bones, solutions to Race to the Gate & Tricky Logic Game

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

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