The caravan, or “What numbers can be covered by 1, 2, 3, 4, 5 and 6?”

So tomorrow is the big day. I’m turning 40 – halfway through the life and towards ending my “career”… From this post, you probably also realize it is vacation times at the moment.

The Caravan

Some years ago, my children got a birthday caravan. The caravan – sold by Goki – consisting of an elephant, giraffe, rhino, camel, crocodile, zebra, goat, and a bear (?), brings candles, gifts, and … a number. The number delivered to us was 1 to 6. Each time we celebrate the childrens’ birthdays, we bring out the caravan and change the number accordingly.

goki_caravan_train

The Goki Birthday Caravan Train

Since you are so clever readers, you quickly realize that we can only celebrate up to 6 years! Luckily, though, the gifts can be replaced by numbers and we can add a combination of numbers, for example 2 and 6 to celebrate our son’s eighth birthday. Phew! What a luck. (Now, it should be mentioned that Goki sells additional numbers up to 10 which would have saved us a lot of time in this post).

However, can we create the number 40 to celebrate my birthday tomorrow?

Let us start in the other end… with the numbers 1 to 6, we can start with the low hanging fruits. Given that we read them in the familiar way, like 12, 13, 14, etc., we can create the following:


1 -- 21 31 41 51 61
2 12 -- 32 42 52 62
3 13 23 -- 43 53 63
4 14 24 34 -- 54 64
5 15 25 35 45 -- 65
6 16 26 36 46 56 --

which keeps us busy until retirement, approximately.

There are obviously numbers missing in this scheme. So, let us now use the trick-that-made-my-son-happy. Let’s assume we can add two numbers together. Can we fill the gaps?


1+2=3
1+3=4 2+3=5
1+4=5 2+4=6 3+4=7
1+5=6 2+5=7 3+5=8 4+5=9
1+6=7 2+6=8 3+6=9 4+6=10 5+6=11

That helped us out a bit, let’s fill out the table and let’s aim for the 99th birthday this time.


0 10 -- -- -- -- -- -- -- --
1 11 21 31 41 51 61 -- -- --
2 12 -- 32 42 52 62 -- -- --
3 13 23 -- 43 53 63 -- -- --
4 14 24 34 -- 54 64 -- -- --
5 15 25 35 45 -- 65 -- -- --
6 16 26 36 46 56 -- -- -- --
7 -- -- -- -- -- -- -- -- --
8 -- -- -- -- -- -- -- -- --
9 -- -- -- -- -- -- -- -- --

Hmm, better, but obviously, someone is going to be upset when they turn 17, just about to enter adulthood and they cannot see their caravan on the birthday table… th th th! Can we add three numbers? Let’s see!


1+2+3=6
1+2+4=7 2+3+4=9
1+2+5=8 2+3+5=10 3+4+5=12
1+2+6=9 2+3+6=11 3+4+6=13 4+5+6=15

Not much luck there 😦 With four numbers it is easier… we obviously see that we lack the number 17, and if we take 6+5+4+3 it is 18 and thus we should be able to cover both 17 and 18 by adding four numbers. With five numbers we can reach 6+5+4+3+2 = 20 . Six numbers doesn’t add value, since we have 21 already catered for.


0 10 20 -- -- -- -- -- -- --
1 11 21 31 41 51 61 -- -- --
2 12 -- 32 42 52 62 -- -- --
3 13 23 -- 43 53 63 -- -- --
4 14 24 34 -- 54 64 -- -- --
5 15 25 35 45 -- 65 -- -- --
6 16 26 36 46 56 -- -- -- --
7 17 -- -- -- -- -- -- -- --
8 18 -- -- -- -- -- -- -- --
9 19 -- -- -- -- -- -- -- --

Still work to do. First, however, since we now can use the addition, we can for example create 27 as 26 + 1 = 27 . Let’s fill out those too.


0 10 20 -- -- -- -- -- -- --
1 11 21 31 41 51 61 -- -- --
2 12 -- 32 42 52 62 -- -- --
3 13 23 33 43 53 63 -- -- --
4 14 24 34 44 54 64 -- -- --
5 15 25 35 45 55 65 -- -- --
6 16 26 36 46 56 66 -- -- --
7 17 27 37 47 57 67 -- -- --
8 18 28 38 48 58 68 -- -- --
9 19 29 39 49 59 69 -- -- --

Seems like we have no other option than to invoke Mr. Multiplication (remember that we have options to let the different gifts act as multiplication and addition sign!) I know I was “cheating” before as we in fact did multiplication by 10 and addition to obtain 24, 25, etc., … maybe we can come back to that in another post… .

Multiplying all numbers gives us a biblical age of 1 x 2 x 3 x 4 x 5 x 6 = 720, which normally is a bit excessive, unless you’re an oak or so. Let’s us see what gaps in the table we can fill, just by a crude approach: 22 = 2 x (5 + 6), 30 = 3 x (4 + 6), 40 = 5 x (6 + 2), Yippiieeee! I will get a caravan train tomorrow! , 50 = 5 x (4 + 6). Then, since 6 x (5+4) < 60, we have to go wild to reach 60: 60 = 6 x (5 + 3 + 2).

Now, all numbers up to 69 are covered – and beyond that, perhaps we do not even want to celebrate…


0 10 20 30 40 50 60 -- -- --
1 11 21 31 41 51 61 -- -- --
2 12 22 32 42 52 62 -- -- --
3 13 23 33 43 53 63 -- -- --
4 14 24 34 44 54 64 -- -- --
5 15 25 35 45 55 65 -- -- --
6 16 26 36 46 56 66 -- -- --
7 17 27 37 47 57 67 -- -- --
8 18 28 38 48 58 68 -- -- --
9 19 29 39 49 59 69 -- -- --

“Concludingly”

All, right so with some twist we can at least create the numbers from 1 to 69 which will keep us busy a while. However, back to the original question:

Assume we have 1, 2, 3, 4, 5, 6 and “+” and “*” available – what numbers can we create?

and what if we are not allowed to use any parentheses? Etc., etc. A topic for discussion and another post I guess.

numbers

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13 thoughts on “The caravan, or “What numbers can be covered by 1, 2, 3, 4, 5 and 6?”

  1. According to my calculation there can be 244176 different numbers possible with 1, 2, 3, 4, 5, 6 and “+” and “*” available.

    Here’s a subset (length 6338): which consists of 1956 possible combinations of the six numbers without any operation plus just one addition and multiplication operation.

    Happy Birthday 🙂

    [1, 2, 3, 4, 5, 6, 12, 13, 14, 15, 16, 21, 23, 24, 25, 26, 31, 32, 34, 35, 36, 41, 42, 43, 45, 46, 51, 52, 53, 54, 56, 61, 62, 63, 64, 65, 123, 124, 125, 126, 132, 134, 135, 136, 142, 143, 145, 146, 152, 153, 154, 156, 162, 163, 164, 165, 213, 214, 215, 216, 231, 234, 235, 236, 241, 243, 245, 246, 251, 253, 254, 256, 261, 263, 264, 265, 312, 314, 315, 316, 321, 324, 325, 326, 341, 342, 345, 346, 351, 352, 354, 356, 361, 362, 364, 365, 412, 413, 415, 416, 421, 423, 425, 426, 431, 432, 435, 436, 451, 452, 453, 456, 461, 462, 463, 465, 512, 513, 514, 516, 521, 523, 524, 526, 531, 532, 534, 536, 541, 542, 543, 546, 561, 562, 563, 564, 612, 613, 614, 615, 621, 623, 624, 625, 631, 632, 634, 635, 641, 642, 643, 645, 651, 652, 653, 654, 1234, 1235, 1236, 1243, 1245, 1246, 1253, 1254, 1256, 1263, 1264, 1265, 1324, 1325, 1326, 1342, 1345, 1346, 1352, 1354, 1356, 1362, 1364, 1365, 1423, 1425, 1426, 1432, 1435, 1436, 1452, 1453, 1456, 1462, 1463, 1465, 1523, 1524, 1526, 1532, 1534, 1536, 1542, 1543, 1546, 1562, 1563, 1564, 1623, 1624, 1625, 1632, 1634, 1635, 1642, 1643, 1645, 1652, 1653, 1654, 2134, 2135, 2136, 2143, 2145, 2146, 2153, 2154, 2156, 2163, 2164, 2165, 2314, 2315, 2316, 2341, 2345, 2346, 2351, 2354, 2356, 2361, 2364, 2365, 2413, 2415, 2416, 2431, 2435, 2436, 2451, 2453, 2456, 2461, 2463, 2465, 2513, 2514, 2516, 2531, 2534, 2536, 2541, 2543, 2546, 2561, 2563, 2564, 2613, 2614, 2615, 2631, 2634, 2635, 2641, 2643, 2645, 2651, 2653, 2654, 3124, 3125, 3126, 3142, 3145, 3146, 3152, 3154, 3156, 3162, 3164, 3165, 3214, 3215, 3216, 3241, 3245, 3246, 3251, 3254, 3256, 3261, 3264, 3265, 3412, 3415, 3416, 3421, 3425, 3426, 3451, 3452, 3456, 3461, 3462, 3465, 3512, 3514, 3516, 3521, 3524, 3526, 3541, 3542, 3546, 3561, 3562, 3564, 3612, 3614, 3615, 3621, 3624, 3625, 3641, 3642, 3645, 3651, 3652, 3654, 4123, 4125, 4126, 4132, 4135, 4136, 4152, 4153, 4156, 4162, 4163, 4165, 4213, 4215, 4216, 4231, 4235, 4236, 4251, 4253, 4256, 4261, 4263, 4265, 4312, 4315, 4316, 4321, 4325, 4326, 4351, 4352, 4356, 4361, 4362, 4365, 4512, 4513, 4516, 4521, 4523, 4526, 4531, 4532, 4536, 4561, 4562, 4563, 4612, 4613, 4615, 4621, 4623, 4625, 4631, 4632, 4635, 4651, 4652, 4653, 5123, 5124, 5126, 5132, 5134, 5136, 5142, 5143, 5146, 5162, 5163, 5164, 5213, 5214, 5216, 5231, 5234, 5236, 5241, 5243, 5246, 5261, 5263, 5264, 5312, 5314, 5316, 5321, 5324, 5326, 5341, 5342, 5346, 5361, 5362, 5364, 5412, 5413, 5416, 5421, 5423, 5426, 5431, 5432, 5436, 5461, 5462, 5463, 5612, 5613, 5614, 5621, 5623, 5624, 5631, 5632, 5634, 5641, 5642, 5643, 6123, 6124, 6125, 6132, 6134, 6135, 6142, 6143, 6145, 6152, 6153, 6154, 6213, 6214, 6215, 6231, 6234, 6235, 6241, 6243, 6245, 6251, 6253, 6254, 6312, 6314, 6315, 6321, 6324, 6325, 6341, 6342, 6345, 6351, 6352, 6354, 6412, 6413, 6415, 6421, 6423, 6425, 6431, 6432, 6435, 6451, 6452, 6453, 6512, 6513, 6514, 6521, 6523, 6524, 6531, 6532, 6534, 6541, 6542, 6543, 12345, 12346, 12354, 12356, 12364, 12365, 12435, 12436, 12453, 12456, 12463, 12465, 12534, 12536, 12543, 12546, 12563, 12564, 12634, 12635, 12643, 12645, 12653, 12654, 13245, 13246, 13254, 13256, 13264, 13265, 13425, 13426, 13452, 13456, 13462, 13465, 13524, 13526, 13542, 13546, 13562, 13564, 13624, 13625, 13642, 13645, 13652, 13654, 14235, 14236, 14253, 14256, 14263, 14265, 14325, 14326, 14352, 14356, 14362, 14365, 14523, 14526, 14532, 14536, 14562, 14563, 14623, 14625, 14632, 14635, 14652, 14653, 15234, 15236, 15243, 15246, 15263, 15264, 15324, 15326, 15342, 15346, 15362, 15364, 15423, 15426, 15432, 15436, 15462, 15463, 15623, 15624, 15632, 15634, 15642, 15643, 16234, 16235, 16243, 16245, 16253, 16254, 16324, 16325, 16342, 16345, 16352, 16354, 16423, 16425, 16432, 16435, 16452, 16453, 16523, 16524, 16532, 16534, 16542, 16543, 21345, 21346, 21354, 21356, 21364, 21365, 21435, 21436, 21453, 21456, 21463, 21465, 21534, 21536, 21543, 21546, 21563, 21564, 21634, 21635, 21643, 21645, 21653, 21654, 23145, 23146, 23154, 23156, 23164, 23165, 23415, 23416, 23451, 23456, 23461, 23465, 23514, 23516, 23541, 23546, 23561, 23564, 23614, 23615, 23641, 23645, 23651, 23654, 24135, 24136, 24153, 24156, 24163, 24165, 24315, 24316, 24351, 24356, 24361, 24365, 24513, 24516, 24531, 24536, 24561, 24563, 24613, 24615, 24631, 24635, 24651, 24653, 25134, 25136, 25143, 25146, 25163, 25164, 25314, 25316, 25341, 25346, 25361, 25364, 25413, 25416, 25431, 25436, 25461, 25463, 25613, 25614, 25631, 25634, 25641, 25643, 26134, 26135, 26143, 26145, 26153, 26154, 26314, 26315, 26341, 26345, 26351, 26354, 26413, 26415, 26431, 26435, 26451, 26453, 26513, 26514, 26531, 26534, 26541, 26543, 31245, 31246, 31254, 31256, 31264, 31265, 31425, 31426, 31452, 31456, 31462, 31465, 31524, 31526, 31542, 31546, 31562, 31564, 31624, 31625, 31642, 31645, 31652, 31654, 32145, 32146, 32154, 32156, 32164, 32165, 32415, 32416, 32451, 32456, 32461, 32465, 32514, 32516, 32541, 32546, 32561, 32564, 32614, 32615, 32641, 32645, 32651, 32654, 34125, 34126, 34152, 34156, 34162, 34165, 34215, 34216, 34251, 34256, 34261, 34265, 34512, 34516, 34521, 34526, 34561, 34562, 34612, 34615, 34621, 34625, 34651, 34652, 35124, 35126, 35142, 35146, 35162, 35164, 35214, 35216, 35241, 35246, 35261, 35264, 35412, 35416, 35421, 35426, 35461, 35462, 35612, 35614, 35621, 35624, 35641, 35642, 36124, 36125, 36142, 36145, 36152, 36154, 36214, 36215, 36241, 36245, 36251, 36254, 36412, 36415, 36421, 36425, 36451, 36452, 36512, 36514, 36521, 36524, 36541, 36542, 41235, 41236, 41253, 41256, 41263, 41265, 41325, 41326, 41352, 41356, 41362, 41365, 41523, 41526, 41532, 41536, 41562, 41563, 41623, 41625, 41632, 41635, 41652, 41653, 42135, 42136, 42153, 42156, 42163, 42165, 42315, 42316, 42351, 42356, 42361, 42365, 42513, 42516, 42531, 42536, 42561, 42563, 42613, 42615, 42631, 42635, 42651, 42653, 43125, 43126, 43152, 43156, 43162, 43165, 43215, 43216, 43251, 43256, 43261, 43265, 43512, 43516, 43521, 43526, 43561, 43562, 43612, 43615, 43621, 43625, 43651, 43652, 45123, 45126, 45132, 45136, 45162, 45163, 45213, 45216, 45231, 45236, 45261, 45263, 45312, 45316, 45321, 45326, 45361, 45362, 45612, 45613, 45621, 45623, 45631, 45632, 46123, 46125, 46132, 46135, 46152, 46153, 46213, 46215, 46231, 46235, 46251, 46253, 46312, 46315, 46321, 46325, 46351, 46352, 46512, 46513, 46521, 46523, 46531, 46532, 51234, 51236, 51243, 51246, 51263, 51264, 51324, 51326, 51342, 51346, 51362, 51364, 51423, 51426, 51432, 51436, 51462, 51463, 51623, 51624, 51632, 51634, 51642, 51643, 52134, 52136, 52143, 52146, 52163, 52164, 52314, 52316, 52341, 52346, 52361, 52364, 52413, 52416, 52431, 52436, 52461, 52463, 52613, 52614, 52631, 52634, 52641, 52643, 53124, 53126, 53142, 53146, 53162, 53164, 53214, 53216, 53241, 53246, 53261, 53264, 53412, 53416, 53421, 53426, 53461, 53462, 53612, 53614, 53621, 53624, 53641, 53642, 54123, 54126, 54132, 54136, 54162, 54163, 54213, 54216, 54231, 54236, 54261, 54263, 54312, 54316, 54321, 54326, 54361, 54362, 54612, 54613, 54621, 54623, 54631, 54632, 56123, 56124, 56132, 56134, 56142, 56143, 56213, 56214, 56231, 56234, 56241, 56243, 56312, 56314, 56321, 56324, 56341, 56342, 56412, 56413, 56421, 56423, 56431, 56432, 61234, 61235, 61243, 61245, 61253, 61254, 61324, 61325, 61342, 61345, 61352, 61354, 61423, 61425, 61432, 61435, 61452, 61453, 61523, 61524, 61532, 61534, 61542, 61543, 62134, 62135, 62143, 62145, 62153, 62154, 62314, 62315, 62341, 62345, 62351, 62354, 62413, 62415, 62431, 62435, 62451, 62453, 62513, 62514, 62531, 62534, 62541, 62543, 63124, 63125, 63142, 63145, 63152, 63154, 63214, 63215, 63241, 63245, 63251, 63254, 63412, 63415, 63421, 63425, 63451, 63452, 63512, 63514, 63521, 63524, 63541, 63542, 64123, 64125, 64132, 64135, 64152, 64153, 64213, 64215, 64231, 64235, 64251, 64253, 64312, 64315, 64321, 64325, 64351, 64352, 64512, 64513, 64521, 64523, 64531, 64532, 65123, 65124, 65132, 65134, 65142, 65143, 65213, 65214, 65231, 65234, 65241, 65243, 65312, 65314, 65321, 65324, 65341, 65342, 65412, 65413, 65421, 65423, 65431, 65432, 123456, 123465, 123546, 123564, 123645, 123654, 124356, 124365, 124536, 124563, 124635, 124653, 125346, 125364, 125436, 125463, 125634, 125643, 126345, 126354, 126435, 126453, 126534, 126543, 132456, 132465, 132546, 132564, 132645, 132654, 134256, 134265, 134526, 134562, 134625, 134652, 135246, 135264, 135426, 135462, 135624, 135642, 136245, 136254, 136425, 136452, 136524, 136542, 142356, 142365, 142536, 142563, 142635, 142653, 143256, 143265, 143526, 143562, 143625, 143652, 145236, 145263, 145326, 145362, 145623, 145632, 146235, 146253, 146325, 146352, 146523, 146532, 152346, 152364, 152436, 152463, 152634, 152643, 153246, 153264, 153426, 153462, 153624, 153642, 154236, 154263, 154326, 154362, 154623, 154632, 156234, 156243, 156324, 156342, 156423, 156432, 162345, 162354, 162435, 162453, 162534, 162543, 163245, 163254, 163425, 163452, 163524, 163542, 164235, 164253, 164325, 164352, 164523, 164532, 165234, 165243, 165324, 165342, 165423, 165432, 213456, 213465, 213546, 213564, 213645, 213654, 214356, 214365, 214536, 214563, 214635, 214653, 215346, 215364, 215436, 215463, 215634, 215643, 216345, 216354, 216435, 216453, 216534, 216543, 231456, 231465, 231546, 231564, 231645, 231654, 234156, 234165, 234516, 234561, 234615, 234651, 235146, 235164, 235416, 235461, 235614, 235641, 236145, 236154, 236415, 236451, 236514, 236541, 241356, 241365, 241536, 241563, 241635, 241653, 243156, 243165, 243516, 243561, 243615, 243651, 245136, 245163, 245316, 245361, 245613, 245631, 246135, 246153, 246315, 246351, 246513, 246531, 251346, 251364, 251436, 251463, 251634, 251643, 253146, 253164, 253416, 253461, 253614, 253641, 254136, 254163, 254316, 254361, 254613, 254631, 256134, 256143, 256314, 256341, 256413, 256431, 261345, 261354, 261435, 261453, 261534, 261543, 263145, 263154, 263415, 263451, 263514, 263541, 264135, 264153, 264315, 264351, 264513, 264531, 265134, 265143, 265314, 265341, 265413, 265431, 312456, 312465, 312546, 312564, 312645, 312654, 314256, 314265, 314526, 314562, 314625, 314652, 315246, 315264, 315426, 315462, 315624, 315642, 316245, 316254, 316425, 316452, 316524, 316542, 321456, 321465, 321546, 321564, 321645, 321654, 324156, 324165, 324516, 324561, 324615, 324651, 325146, 325164, 325416, 325461, 325614, 325641, 326145, 326154, 326415, 326451, 326514, 326541, 341256, 341265, 341526, 341562, 341625, 341652, 342156, 342165, 342516, 342561, 342615, 342651, 345126, 345162, 345216, 345261, 345612, 345621, 346125, 346152, 346215, 346251, 346512, 346521, 351246, 351264, 351426, 351462, 351624, 351642, 352146, 352164, 352416, 352461, 352614, 352641, 354126, 354162, 354216, 354261, 354612, 354621, 356124, 356142, 356214, 356241, 356412, 356421, 361245, 361254, 361425, 361452, 361524, 361542, 362145, 362154, 362415, 362451, 362514, 362541, 364125, 364152, 364215, 364251, 364512, 364521, 365124, 365142, 365214, 365241, 365412, 365421, 412356, 412365, 412536, 412563, 412635, 412653, 413256, 413265, 413526, 413562, 413625, 413652, 415236, 415263, 415326, 415362, 415623, 415632, 416235, 416253, 416325, 416352, 416523, 416532, 421356, 421365, 421536, 421563, 421635, 421653, 423156, 423165, 423516, 423561, 423615, 423651, 425136, 425163, 425316, 425361, 425613, 425631, 426135, 426153, 426315, 426351, 426513, 426531, 431256, 431265, 431526, 431562, 431625, 431652, 432156, 432165, 432516, 432561, 432615, 432651, 435126, 435162, 435216, 435261, 435612, 435621, 436125, 436152, 436215, 436251, 436512, 436521, 451236, 451263, 451326, 451362, 451623, 451632, 452136, 452163, 452316, 452361, 452613, 452631, 453126, 453162, 453216, 453261, 453612, 453621, 456123, 456132, 456213, 456231, 456312, 456321, 461235, 461253, 461325, 461352, 461523, 461532, 462135, 462153, 462315, 462351, 462513, 462531, 463125, 463152, 463215, 463251, 463512, 463521, 465123, 465132, 465213, 465231, 465312, 465321, 512346, 512364, 512436, 512463, 512634, 512643, 513246, 513264, 513426, 513462, 513624, 513642, 514236, 514263, 514326, 514362, 514623, 514632, 516234, 516243, 516324, 516342, 516423, 516432, 521346, 521364, 521436, 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    • Aah! Great.

      I saw the python script on your philectronics blog.

      Before i hack: Are all numbers 1 to 100 catered for if one sorts and uniquifies this list?

      • 2*(5+6) involves 2 operations, but that script gives:
        numbers formed by different combinations of the 6 numbers ex:- 3, 25, 631,etc
        and
        numbers formed by just one operation (either + or *) ex :- 2*5, 12+34, 123*456

  2. Pingback: An Answer to : The caravan, or “What numbers can be covered by 1, 2, 3, 4, 5 and 6?” | philectronics

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