Ask The Wizard #297

What was that last play in the first half of the Panthers/Saints game in week 13 of the 2015 season?


I believe it is referred to as a "point after touchdown return." Per a rule change effective this season, if an attempted kick after a touchdown fails, with the opposing team scoring a touchdown, then it shall count as two points for the scoring team. Prior to the 2015 season, it would have been a dead ball. This was the first, and so far only, such score in NFL. You can see a video of it at YouTube.

This question was raised and discussed in my forum at Wizard of Vegas.

What is your opinion of the player who was denied an $8.5 million slot machine jackpot at the Lucky Eagle Casino in Rochester, Wash. Seems she has a good case to me.

terapined from Tampa, FL

She was playing a 5¢ machine and the alleged win was $8,588,749.65. This number looked familiar to me as a power of 2. To be specific, 2^33 = 8,589,934,592. If this were a count of tenths of pennies, it would equal $8,589,934.592. The difference between that and the jackpot is $1,184.942.

What I think happened is the machine declared the win as an unsigned integer, meaning a number that could never go negative. However, through some programming error, it wanted to. When you try to put a negative number into an unsigned integer, the computer will wrap around the other end. In this case, I think something bizarre happened and through whatever programming error, the game thought the player had a loss of $1,184.942. When it tried to display this number as an unsigned integer, it wrapped around the maximum value and displayed the win of $8,588,749.65.

Every slot machine I've ever seen says somewhere "malfunction voids all pays and plays." If I were the judge, I would have to say that this was indeed a malfunction and side with the casino. This was, of course, what the casino argued. Nevertheless, their $80 compensation offer strikes me as very stingy.

More information at KATU.

For more discussion on this question, please see my forum at Wizard of Vegas.

Love your site. I have just been to Panama (in fact I'm here now) and I have some very bad news. As of 2015, all Panamanian casinos must charge the customer exactly 5.5% commission on all chips that are changed at the cage no matter if the player wins or loses. I was told that it is to pay for the retirees and future retirees in the country. Why they don't just tax the casinos instead of the winning players is beyond me. I figure they are trying to reduce the amount that the locals gamble. Never have I seen a Latin American city with so many casinos. With a tear in my eye, I walked from one to another and the story was the same. For any serious gambler, Panama City is dead to us as well as the entire isthmus. What a pity.

Mexico is the same, but has always been that way. Just for fun, you should list the player's expected return in these countries.


Thank you for the compliment. That is really sad. This new law evidently took effect around July, 2015. No longer will I patronize my favorite casino in the world -- the Venato in Panama City. As far as my Latin American gambling is concerned, it is good-bye Panama, hello Costa Rica.

Regarding Mexico, you're the second person to comment to me this week about an alleged transaction tax or win tax in Mexico. I can't speak for the whole country, but I played at the King's Casino in Mexico City last month, won, and was paid in full.

Regarding listing expected returns by country, there is no way to mathematically convert a chip tax upon cash-in and house edge. The effect of this Panamanian tax would largely depend on how long the player circulates through his chips, the more he does, the less the effect. My advice to players, if they must play, would be to buy in for very small amounts and then flat bet. If you bust out, then make another small buy in. If you walk away from the table with chips, don't cash them in unless there is no chance you'll ever return to the tables. If you're with friends, try to buy chips from each other, rather than the casino, so the seller will get full value.

For more information, I recommend The article Tax confusion hits Panama's casino sector.

A field of grass can feed exactly:

One cow and one llama for 21 days.
One llama and one sheep for 42 days.
One sheep and one cow for 28 days.
The cow eats as much grass as the llama and the sheep together.
The grass grows at a constant rate.

How long will it take the three animals together to totally devour the field of grass?


c = rate cow eats grass
l = rate llama eats grass
s = rate sheep eats grass
g = rate grass grows

At the end of a period of time, grass consumed must equal the initial amount of grass plus the amount of grass grown in that time. So...

(1) 21*(c+l) = 1 + 21g
(2) 42*(l+s) = 1+42g
(3) 28*(s+c) = 1+28g

Where the 1 represents one field of grass.

We are also given:

(4) c=s+l

First, substitute equation (4) into (2):

(5) 42c = 1 + 42g

Express that in terms of g:

(6) g = (42c-1)/42

Next, substitute equation (6) into (1)...

(7) 21(c+l) = 1 + 21*(42c-1)/42

After a little algebra we get...

(8) l = 1/42.

Next, substitute equation (4) into (3)...

(9) 28*(2s + l) = 1+28g

We know l=1/42, so...

28*(2s + 1/42) = 1+28g
56s + 28/42 = 1 + 28g
2352s + 28 = 42 + 1176g
(10) g = (2352s - 14)/1176

Next, substitute equation (8) and (10) into (2) ...

42*(1/42 + s) = 1 + 42*(2352s - 14)/1176

After some easy algebra we get:

(11) s = 14/1176 = 1/84

From equation (4)

(12) c = (1/84) + (1/42) = 3/84 = 1/28

So, if the grass didn't grow, then it would take the cow 28 days to eat the field, the llama 42, and the sheep 84.

Next, let's solve for g. Substitute (11) into (10):

g = [2352*(1/84)- 14]/1176
(13) g = 14/1176 = 1/84.

This is coincidentally the same rate the sheep eats the grass.

Let t be the final answer. We know that in t days the amount of grass eaten must equal the amount of grass in the field (1) plus the grass grown in that time. So...

(13) t*(s+l+c) = 1 + tg

Solving for t...

t*[(1/84) + (1/42) + (1/28)] = 1 + t/84
t = 1/[(1/84) + (1/42) + (1/28) - (1/84)]
(14) t = 84/5 = 16.8 days = 16 days, 19 hours, 12 minutes

This question was raised and discussed in my forum at Wizard of Vegas.