Below is the Fox Business holiday card, clearly designed by the same “artist” who did the Fox News card, with the same creepily aggressive tone, and similarly depicting the audience as sheep.

# We have a James Carville / Mary Matalin thing going on around here

Among the holiday cards we received this year was the following classy item from the Obama family,

Lovely, though the embossed gold presidential seal seems a little over the top — was anybody really going to be unclear about the card’s origin? “Hey honey, is this from the Barack Obama at the dry cleaners, or the one who married your cousin Lois?”

We also got the following, oddly aggressive card from Fox News,

I’m surprised that the graphics department at Fox couldn’t do any better, and I’ve never before seen holiday cards used to mock one’s professional rivals. Most disturbing — they depict their own audience as sheep.

To their credit, Fox included a really nice game set (playing cards, dice, checkers, etc.) as their holiday gift.

# Sled Dogs in Vermont

I brought the snow skates up to Vermont last week, and nobody recognized them at all. Typically some old-timers will remember them from the 1990s, but I guess Sled Dogs never penetrated the Vermont market even when they were still being sold. Virtually everybody assumed I was descending the mountain in ski boots, and they remained impressed even after I explained that the reality was somewhat less insane than that.

We got about 18 inches of snow one night, giving me the opportunity to re-learn just how worthless these things are in deep powder.

In related news, the company that currently owns the rights to the sled dog name and design has given up on their plan to re-enter the U.S. market this season.

“Sled Dogs snowskates will sadly not be available this season. We thought it would, but due to quality issues with our production, we have to postpone the re-launch to next season. We are not happy about this and understand that our customers are the same. However we can not sell products that do not have the quality that is up to our standards. We will re-launch the webpage when we have products of right quality in stock in our warehouse. We do not know now when this will be but most likely it will be for next season.”

# A perfect solution to 24s

In several recent posts, I have discussed a method for perfect solution to the game of 24s. Here, I summarize some of those results and link to the Mathematica notebook I used to find them, and a comprehensive list of solutions for those who would like to skip the math and get right to the answer.

To recap, there are 715 possible hands in a game of 24s. 566 of those hands can be solved with arithmetics alone, 595 if we add roots, powers, and logs, and 619 if we add Mod.

I generally play with Log and Power and Root allowed, but not Mod. I find that the hands that are solvable only with Log, Power, and Root, generally are solvable by humans. See for example

{1,1,8,9} -> 8 Root[9,1+1]

{1,2,9,9} -> (9-1) Root[9,2]

{2, 2, 2, 6} -> 2^(2+Log[2,6]) or (2*Root[6,2])^2

{2,2,9,9} -> 2(Root[9,2]+9)

{3,7,8,10} -> 7 Root[8,3] +10

{3,4,9,10} -> 4+Log[3,9]*10

{7, 8, 9, 9} -> 8 Root[9,9-7], and

{8, 8, 9, 10} -> 8 Root[9,10-8].

A bright high school student should be able to find any of those. However, Mod introduces solutions that would probably never be played by an unaided human being. For example,

{5, 6, 7, 10} -> 6*Mod[7^10,5], and

{5,7,7,8} -> 8*Mod[7^7,5].

There is no other solution in these two cases.

Anyway, I’ve provided a comprehensive list of solutions linked as RTF and PDF documents, and the Mathematica notebook that I used to solve the problem in the first place.

And, most importantly, the Mathematica notebook.

Let me know if you do anything interesting with this.

# Solving 24s, part iii: mod

I have shown how to comprehensively solve a game of 24s that allows just arithmetic, and how to expand that to allow powers, roots, and logarithms. Now we’re going to add the modulo (Mod) function.

If you’ve followed the methodology in parts i and ii, this is pretty simple stuff.

First, to our set of simplified function definitions, I add

```
mod[x_Integer?(# > 0 &), y_Integer?(-16 <= # <= 16 && # != 0 &)] := Mod[x, y];
mod[x__] := Indeterminate;
```

Then, our rule definition step looks like

`rules = generateRules[{Times, Plus, Subtract, divide, power, root, log, mod}, 4];`

We're up to 512 rules here, once redundant patterns are eliminated. The complexity increases exponentially with the number of operators allowed.

The good news is that Mod makes lots of hands solvable that wouldn't be otherwise. See, for example,

{3, 5, 5, 10} -> 5*5-Mod[10,3]

{5, 6, 7, 10} -> 6*Mod[7^10,5]

{5, 7, 7, 8} -> Mod[7^7,5]*8, and

{6, 7, 7, 9} -> 6*Mod[7*7,9]

In fact, allowing Mod as an operator increases the total number of solvable hands all the way to 619 out of 715.

The unsolvable hands that remain are

{{1, 1, 1, 1}, {1, 1, 1, 2}, {1, 1, 1, 3}, {1, 1, 1, 4}, {1, 1, 1, 6}, {1, 1, 1, 7}, {1, 1, 1, 9}, {1, 1, 1, 10}, {1, 1, 2, 2}, {1, 1, 2, 3}, {1, 1, 5, 9}, {1, 1, 5, 10}, {1, 1, 6, 7}, {1, 1, 6, 10}, {1, 1, 7, 7}, {1, 1, 7, 8}, {1, 1, 7, 9}, {1, 1, 8, 10}, {1, 1, 9, 9}, {1, 1, 9, 10}, {1, 1, 10, 10}, {1, 2, 2, 2}, {1, 2, 9, 10}, {1, 2, 10, 10}, {1, 4, 9, 9}, {1, 5, 7, 7}, {1, 6, 6, 7}, {1, 6, 7, 7}, {1, 6, 7, 8}, {1, 6, 10, 10}, {1, 7, 7, 7}, {1, 7, 7, 8}, {1, 7, 10, 10}, {1, 8, 9, 9}, {1, 8, 9, 10}, {1, 8, 10, 10}, {1, 9, 9, 9}, {1, 9, 9, 10}, {1, 9, 10, 10}, {1, 10, 10, 10}, {2, 2, 2, 2}, {2, 2, 7, 9}, {2, 6, 7, 7}, {2, 7, 7, 7}, {2, 7, 7, 9}, {2, 7, 9, 9}, {2, 9, 9, 9}, {2, 9, 9, 10}, {2, 10, 10, 10}, {3, 3, 10, 10}, {3, 5, 7, 7}, {3, 10, 10, 10}, {4, 4, 9, 9}, {4, 7, 7, 9}, {4, 7, 7, 10}, {4, 9, 9, 9}, {5, 5, 5, 7}, {5, 5, 5, 8}, {5, 5, 5, 10}, {5, 5, 6, 10}, {5, 5, 7, 9}, {5, 7, 7, 7}, {5, 7, 9, 9}, {5, 8, 10, 10}, {5, 9, 9, 9}, {5, 10, 10, 10}, {6, 6, 6, 7}, {6, 6, 7, 7}, {6, 6, 7, 8}, {6, 7, 7, 7}, {6, 7, 7, 8}, {6, 9, 10, 10}, {7, 7, 7, 7}, {7, 7, 7, 8}, {7, 7, 7, 9}, {7, 7, 7, 10}, {7, 7, 8, 8}, {7, 7, 8, 9}, {7, 7, 9, 9}, {7, 8, 8, 8}, {7, 9, 9, 9}, {7, 9, 9, 10}, {7, 9, 10, 10}, {7, 10, 10, 10}, {8, 8, 8, 8}, {8, 8, 9, 9}, {8, 8, 10, 10}, {8, 9, 9, 9}, {8, 9, 9, 10}, {8, 9, 10, 10}, {8, 10, 10, 10}, {9, 9, 9, 9}, {9, 9, 9, 10}, {9, 9, 10, 10}, {9, 10, 10, 10}, {10, 10, 10, 10}}