Nostalgia

[simont]

Trying to recapture your childhood is always dangerous. Books you read, games you played and TV you watched as a child are all things you can dig out and read, play or watch again; sometimes they'll be as good as you remember, but often they won't.

Usually that's because you have changed, of course; but not in this case. This month I remembered a game my father taught me some time around my late teens: you take the four digits of the current year, and you attempt to combine them arithmetically to form each number from 1 upwards and see how far you can get. You're allowed to add, subtract, multiply and divide, you're allowed to use parentheses (of course), and you're also allowed to start by concatenating some of the digits into larger numbers if you want. The catch is that you have to use all four digits every time; if one or two of them can easily be combined to produce the target you're after, you have to find a way to safely dispose of the others. The next year, you can start all over again and it'll all be completely different.

So in 1992, for example, I might have started with 1=2-1+9-9 and 2=1+2-9/9, got as far as 22=21+9/9, and had trouble with 23. It needn't stop there, of course; I might have skipped 23 and tried for things above that.

Like so many things one remembers fondly from one's childhood, this game is not as much fun as I remember it being; but this time it isn't me who's changed. When two of the digits of the current year are zeroes, it gets very boring! If anyone is contemplating having a go at this game, I urge them to wait until at least 2011; and I don't think the game will really recover all of its fun until some time around 2134.

Comments

[aldabra]

I'm sure I remember a game like that, where all four of your numbers were 7. So you had 77/77, 7/7 + 7/7, (7+7+7)/7, and so on. I think I remember getting to about 23, but it's possible I skipped some.

Perhaps you could use the numbers in 26/10/2005, which looks as if it has an expected yield of slightly under five non-zero numbers. If you excluded duplicates it might work nicely. But my head hasn't started working yet this morning and I reserve the right to be talking cobblers.

[simont]

I've recently written a program which searches for all the numbers you can make from a set of inputs in this way (and is also adaptable to the slightly different rules of the Countdown numbers game). It thinks the possibilities given four 7s start off 1-10, 12-15, 18, 28; so if you got 23 or anything near it then either there's a bug in my program or you weren't playing by exactly the same rules...

It turns out that anyone playing my dad's version in 1872 would have had the most fun of anyone: you can get into the 80s before meeting an impossible number.

I also played a similar game involving four 4s, but that one seemed to have more open-ended rules: you were allowed to do any mathematical operation you could think up which didn't need a number to be written down to express it. So you could raise things to the power 4, you could write √4 any time you needed a 2, and I vaguely remember finding a use for ⌊cosh 4⌋ = 27.

[aldabra]

Oh, I was definitely allowed powers and roots and stuff, and it's possible it was with 4s rather than 7s (roots of 7 being less useful).

[dennyd.livejournal.com]

We did four 4s in one of my maths GCSE classes... I remember it being a lot of fun, and I stayed after school to keep trying to get some of the ones that weren't immediately obvious. My maths teacher was quite pleased :)

[gerald_duck]

This may be one of the best arguments yet for conversion to Judaism; 5766 probably works a whole lot better than 2005.

[stephdairy.livejournal.com]

Or Islam, in which it is 1426.

(S)

[cartesiandaemon.livejournal.com]

Surely you have changed, in that now you could solve/look up a solution to this?

Often as much fun can be found in the meta-game, ie. choosing which operators are least a cheat to use.

[simont]

I'm unconvinced that I wouldn't have been just as able to write a solver in 1992 as I am now. It might have taken me a bit more time, but I don't think my skills would have been inadequate to the task. The only difference is that now I have much faster machines available to run it on (so that, for example, I can determine in under thirty seconds that 1,2,7,8 is the set of four starting digits which renders the longest initial subsequence of ℕ reachable), but that isn't a change in me either :-)

[cartesiandaemon.livejournal.com]

True, but you *didn't*. If you're similar to me, at one point, your first thought was "Ooh, this is fun" and at a later point your first thought is "This is a solved problem. How do I find the general answer with a formula or computer again?"

Both are equally interesting reactions, because the first leads to the second, whilest often *doing* the first leads to more understanding (eg. suggestions for similar games/theorems), but it shows a change in view.

Or am I bullshitting? :)

[simont]

Actually, my primary purpose in writing a solver is to turn it round and use it for generation: my hope is that I'll be able to invent puzzles of this type with vaguely consistent difficulty.

(There are two particularly good ones I know of: try making 24 using 3,3,8,8, and then try making 1 using 1,1,1,5. In both these cases you aren't allowed digit concatenation, and you must use all the numbers.)

[shadowphiar.livejournal.com]

Are those puzzles using the restricted rules, or am I allowed to invoke wacky operators?

[simont]

The two I just mentioned? No wacky operators needed there. Addition, subtraction, multiplication and division only. Evaluation order is unrestricted (i.e. use as many parentheses as you like). No digit concatenation: the four inputs must be used as separate numbers and combined only by arithmetic operations.

[cartesiandaemon.livejournal.com]

Gaaah! I thought "I won't download any excecutables or source, that should keep be safe from simon's insiduous puzzles..." but no, apparently *talking* to you is enough to ruin my productivity :)

I know I've solved the 1115 thing *before*, but totally mind-blanked, and wrote a perl script to try all combinations. Doh!

[ewx.livejournal.com]

1 from 1115 I got reasonably quickly when it came up one evening. 24 from 3388 I'm still thinking on...

[timotab.livejournal.com]

with the year thing, I the rules I had was that the numerals had to be in order.

technically, instead of doing 1=2-1+9-9, you had to do -1+9-9+2, and 2= 1-9/9+2

Made some of the numbers a bit harder.

[xaosenkosmos.livejournal.com]

It also makes for really bad one-liners, like:
"13? Ask me again in a decade or so."

[simont]

That rings a bell, actually. I think I knew of that variant but didn't usually play it.

A quick fiddle with my software and another exhaustive search reveals that the most fun year in which to play that variant will be 3625!

[simont]

Er, most fun four-digit year, that is. If you're permitted a starting digit sequence of arbitrary length, you can arrange for as many consecutive numbers from 1 to be reachable as you like: simply select N 1s, and you'll be able to make everything from 1 up to N at least just by adding together some number of 1s and multiplying by the rest.