Think outside the box and consider extending lines beyond the dots.

The trick is to think outside the box — literally. The four lines must extend beyond the boundary of the 3×3 grid. Starting outside the grid on one side allows each line to pass through three dots, then continue past the edge to set up the next line. Most people assume the lines must stay within the square formed by the dots, but the puzzle never says that.

The product of the number of fingers on the left-hands of every person is zero. It only takes one person to have no fingers on their left hand for the product to be zero, because anything multiplied by zero equals zero.

The sister is alive and so could technically speaking be punished, but although probably not by the laws of any western judicial systems.

The sisters are conjoined twins.

They were two of a set of triplets (or quadruplets, etc.)

Try solving the three Tasmanian camels problem first:

This is not an optical illusion. If you print out the puzzle, cut out the pieces and rearrange them, you get the exact same dilemma.

The gradient of the teal hypotenuse is different than the gradient of the red hypotenuse.

This is a classic economics problem that even economists get wrong!

This is a terrible proposition for both you and me.

From my perspective, if I have a hundred people playing, I am losing 5% each round.

From your perspective, unless you are the lucky ten percent (~13.6%), you are losing 5% each round.

Here is the math behind it.

1. House perspective: expected value

Each round, the player’s bankroll is multiplied by:

Heads: ×1.5
Tails: ×0.6

Expected multiplier:

0.5 × 1.5 + 0.5 × 0.6

= 0.75 + 0.30

= 1.05

So the player gains 5% in expected value per round.

That means the house loses 5% in expected value per round.

After 100 rounds, expected player bankroll is:

$10,000 × 1.05^100 ≈ $1,315,013

So from the house’s perspective, this is a terrible game to offer.

2. Player perspective: compound growth

For the player, expected value is misleading because the whole bankroll is being compounded.

The compound-growth multiplier is the geometric mean:

sqrt(1.5 × 0.6)

= sqrt(0.9)

≈ 0.9487

So the typical player loses:

1 - 0.9487 = 0.0513

or about:

5.13% per round

That is why the player’s typical outcome is bad even though the expected value is positive.

3. After 100 rounds

If the player gets H heads and 100 - H tails, final bankroll is:

$10,000 × 1.5^H × 0.6^(100-H)

To finish ahead, they need:

1.5^H × 0.6^(100-H) > 1

Take logs:

H ln(1.5) + (100 - H) ln(0.6) > 0

Solve:

H > 55.749...

So the player needs at least:

56 heads out of 100

The probability of getting 56 or more heads with a fair coin is:

P(H ≥ 56) ≈ 13.6%

So:

About 13.6% of players win.
About 86.4% of players lose.