In The Lady and the Toaster puzzle, you were placed in a deadly gun battle in the world of Battlestar Galactica. You were pitted against two cyborgs, the Lady (a blonde humanoid cyborg also known as Number Six) and the Toaster (a derisive term for the metallic Centurion). Each of you had to fire one shot in turn. You, the weakest shooter, with an accuracy of 1 in 3, went first. You were to be followed by the lady who had an accuracy of 1 in 2. The third shooter was to be the toaster who was perfectly accurate. Anyone who was shot missed their turn which passed to the next contestant still standing, till only one contestant remained.

The first challenge was to determine your best strategy.

It is tempting in this scenario to aim at your most dangerous opponent – the toaster. Quite a few readers were tripped up by this seemingly obvious choice. But let’s
think about that for a moment. If you did that, and were successful, the lady, a better shot than you, would get the added advantage of getting the first shot to finish you off in your duel. This would diminish
your chances considerably. Clearly, common sense dictates that your best chances would be if you never let either of your opponents get the first shot against you. Andy put it well:

“The optimal strategy is to deliberately miss until only 2 contestants remain, since no one will aim at you as long as there are 3 contestants. This strategy ensures that you get the first
shot in the final duel, giving you the best odds at victory.”

So if you fired harmlessly straight up into the air, the lady would be obligated to take aim at her most dangerous opponent, the toaster. If she succeeded, you would get first shot at her, your one surviving opponent. On the other hand, if she failed (as she would be expected to, 50% of the time), the toaster would have to take her out, as she is his most dangerous opponent. And you would again have first crack against your one surviving opponent, the toaster.

Note that the only difference for you between shooting at the toaster and shooting up into the air on your first attempt is that, if you successfully take out the toaster in the former case, the lady gets first crack at you. This never happens in the latter case. Everything else remains exactly the same. Therefore, based on just this simple reasoning it is clear that missing intentionally is the better bet. You can satisfy yourself that it is also much better than aiming at the lady (which is the dumbest thing you could do, because if you were successful, the toaster would take you out with certainty). Hence your best strategy is to miss intentionally.

The next challenge was to calculate the probabilities of survival for each contestant. Probability calculations are daunting to many people, but let me show you how they can be made simple. First you state plainly what the probabilities for the individual events are. Your hit rate is 1/3, so your miss rate is 2/3; the lady’s hit rate is 1/2, and her miss rate is also 1/2; the toaster’s hit rate is 1 and miss rate is 0. That’s easy enough. Now all you have to do is articulate all the ways a contestant can win in plain and simple English. Then just multiply the probabilities where you use the word “and” and add them where you use the word “or.” That’s all there is to it!

Let’s compute the probabilities for the toaster first. The only way in which the toaster can win is:

The lady misses **and** the toaster hits **and** you miss **and** the toaster hits (All **and**s, so multiply the appropriate hit/miss rates together: 1/2 x 1 x 2/3 x 1 = 1/3 = 33.33%).

Now let’s do it for the lady (for short, let’s use use L for lady, T for toaster and Y for you). L can win by first hitting T (if she misses it’s game over for her) and then hitting Y on her 1st or 2nd or 3rd or 4th… try, as follows:

L hits **and** Y miss **and** L hits [1st try:

1/2 x 2/3 x 1/2 = 1/6]

**or**

L hits **and** Y miss **and** L misses **and** Y miss **and** L hits [2nd try: (1/2 x 2/3 x 1/2) x (2/3 x 1/2) = 1/6 x 1/3]

**or**

L hits **and** Y miss **and** L misses **and** Y miss **and** L misses **and** Y miss **and** L hits [3rd try:
(1/2 x 2/3 x 1/2) x (2/3 x 1/2) x (2/3 x 1/2) = 1/6 x 1/3 x 1/3]

**or**…

(And so on, *ad infinitum*)

Now it seems we are in a soup. We have an infinite number of terms that are joined by **or** so they have to be added together: each time both the lady and you miss
(a chance of 1/2 x 2/3 = 1/3), a new term that is 1/3 as much as the previous one is added. Not to worry, because we are now going to invoke one of the most simple and miraculous formulas in all of mathematics!
Let’s gather the terms together:

1/6 + 1/6 x 1/3 + 1/6 x 1/3 x 1/3… = 1/6 x (1 + 1/3 + 1/3 x 1/3 + 1/3 x 1/3 x 1/3…)

The infinite series in the parentheses is what is known as a convergent
geometric series – every new term multiplies the previous one by a constant ratio *r* that is less than 1 (in this case 1/3). The formula for the sum of this series is the absurdly simple 1/(1-*r*)
which is 1/(2/3) = 3/2, which multiplied by the 1/6 outside the parentheses gives 1/4 or 25%. (Having used the formula 1/(1-r) countless times in my life, I’m still amazed at the infinite amount of work done
by this utterly simple expression.) Note that if the first term is *a* instead of 1, the sum is *a*/(1-*r*): still, extremely simple.

We could calculate your chances by enumerating all the possibilities too, but now there’s no need to. Since one of the three contestants has to remain standing at the end, your chances are 1 – 1/3 – 1/4 = 15/36 or 41.67%. As we had deduced, these are higher than your chances would have been had you aimed at the toaster (13/36 = 36.11%) or at the lady (10/36 = 27.78%).

The challenge question was to determine what your accuracy and that of the lady needed to be to make everybody’s chances equal. Here’s how you do it, in brief:

Since the toaster’s chances remain
1/3, the probability that L misses **and** Y miss is still 1/3 (Lm x Ym = 1/3)

We’ve already determined L’s chances as (Lh x Ym x Lh) times 3/2. Setting this equal to 1/3, we get Lh
x Ym x Lh = 2/9. Dividing this equation by the first one, we get (Lh x Lh)/Lm =2/3. Since Lm is 1-Lh, this simplifies to a quadratic equation for Lh of the form 3x^{2}+2x-2=0, whose solution is (sqrt(7)-1)/3
or 0.5486. Your accuracy can then be easily determined to be 0.2616. This equalizes everyone’s chances. This correct answer was first pointed out by Cww alias Cheng Wang, who works for Mozilla in Mountain
View, CA. Cheng gets the prize for this puzzle — a copy of the book, “The Mathematical Puzzles of Sam Loyd” (edited by Martin Gardner).

It is interesting that if you aim for the toaster first you still get a solution for equal chances, a different one with you needing an accuracy of 0.263 and the lady needing only 0.386. As Tom E. pointed out this shows that the player most affected by your strategy of shooting in the air is the lady. The most likely contestant to win if you aimed at the toaster, she now becomes the one least likely to win. If she were truly humanoid, better avoid her eyes when you miss intentionally: surely they would express a look that could have killed — and that would complicate this game even further!

On a more serious note, D-Ferg explored the solution space for this problem in detail and made a surprising discovery, illustrated with an attractive graph: if both you and the lady are weaker shooters than what was stated in this problem, missing intentionally may not always be the best strategy. And Justin Thyme, as usual, had several entertaining and instructive graphics, including an interactive one. Finally, many readers offered high-quality Spoonerisms: we will consider them in a separate blog, with a consolidated word-puzzle prize.

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