## ________________

Another workout in logic is in store for the math team.

## ________________

### Knight Trap

A knight is placed on an infinite chessboard.

*If it cannot move to a square previously visited, how can it be moved in as few moves as possible so that it has no more moves available?*

### My Number is 136

Ana and Boris play a logic game. They both know that possibly identical positive integers, a and b, are chosen by a third party. Their product is written on Ana’s forehead and their sum is written on Boris’s forehead. Each can see the other’s number but not his or her own. After each sees the number on the other’s forehead, one says to the other, “There is no way you can know your number.” The other responds, “I now know my number is 136.”

*What number is on Ana and what number is on Boris?*

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## Solutions to week 130

The thickness of the shell in The Size of Humanity would be about 9.14002103 ten-thousandths of a millimeter. This is less than a micron (= one millionth of a meter). In Fish Pond, the water level is 8 inches above the platform.

**The Size of Humanity answer explained:**

The surface area of the earth is A = 4πR^{2}, where R = 6371 km. The volume of a shell of small thickness, t, is approximately V = At. In our case V = 7.4 × 10^{9} × 63000 × 10^{-15} km^{3} = .4662 km^{3} so t = .4662/(4πR^{2}) = 9.14002103 x 10^{-4} mm or 9.14002103 ten-thousandths of a millimeter.

**Fish Pond answer explained:**

Define L as the side of each cube, A as the surface area of the pond, and d as its original depth. The volume of water is V = Ad. With some thought, it becomes clear that the first and second blocks must not be completely submerged after they are placed. Placing the third block then submerges all three. Suppose the first block causes the water level to rise by h. By conserving the total volume of water, we have V = (A – L^{2})(d + h) = (A – 2L^{2})(d + h + 12) = A(d + h + 24) – 3L^{3}. Combining these three equations gives L = d + h + 16. The water level above the platform is then d + h + 24 – L = 8 inches.

## Recent Weeks

**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

**Week 127**: Race to the Gate & Tricky Logic Game, solutions to Logical Coffee Order & Here’s Looking at You

**Week 126**: Logical Coffee Order & Here’s Looking at You, solutions to Piece of Pi & Making 17 and 27

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

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