## 24 challenge (new developments!)

Anything goes, but keep it seemly...
If you adopt algebraic field as the basis for equivalent formulas, what about distributivity?
The operation * is distributive over the operation +
For all a, b, c, belonging to F, a * (b + c) = (a * b) + (a * c).
AM

Posts: 5
Joined: 26 May 2006

AM wrote:If you adopt algebraic field as the basis for equivalent formulas, what about distributivity?
The operation * is distributive over the operation +
For all a, b, c, belonging to F, a * (b + c) = (a * b) + (a * c).

We recognise this relationship too, but since for this one the left hand side has 3 variables and the right hand side 4, there are no duplication issues so we don't need to worry about it...
udosuk

Posts: 2698
Joined: 17 July 2005

### Re: 24 challenge

udosuk wrote:Anyone has played the game 24 challenge?

Your're given 4 single digit numbers...

When I was originally taught this game, it was with a deck of cards. As such, we inlcuded the number 10 in our calculations (all face cards were assigned the value 10). Udosuk, would you mind generating your list again but allowing for the number 10?
re'born

Posts: 551
Joined: 31 May 2007

Coincidentally, that's the way I was originally taught too...
Here is the list for candidate 1..10, with "X" replacing 10 (Roman/bowling mood )...

1118: ((1+1)+1)*8
1126: ((1+1)+2)*6
1127: (1+2)*(7+1)
1128: ((1*1)+2)*8
1129: (1+2)*(9-1)
112X: ((1+1)+X)*2
1134: (1+1)*(3*4)
1135: (1+3)*(5+1)
1136: ((1+1)+6)*3
1137: ((1*1)+7)*3
1138: ((3*8)-1)+1
1139: (1+1)*(3+9)
113X: (X-(1+1))*3
1144: ((1+1)+4)*4
1145: ((1*1)+5)*4
1146: ((4*6)-1)+1
1147: (7-(1*1))*4
1148: (8-(1+1))*4
1149: (1-4)*(1-9)
114X: ((1+1)*X)+4
1155: ((5*5)*1)-1
1156: (5-(1*1))*6
1157: (1+1)*(5+7)
1158: (5-(1+1))*8
1166: (6-(1+1))*6
1168: 8/((1+1)/6)
1169: ((1+1)*9)+6
117X: ((1+1)*7)+X
1188: ((1+1)*8)+8
1224: (1+2)*(2*4)
1225: (1+5)*(2+2)
1226: (1+2)*(2+6)
1227: (7-1)*(2+2)
1228: (2-(1-2))*8
1229: ((1+2)+9)*2
122X: (1+2)*(X-2)
1233: (1+3)*(3*2)
1234: ((1+2)+3)*4
1235: ((1+2)+5)*3
1236: (3-(1-2))*6
1237: (1+2)+(3*7)
1238: (2-1)*(3*8)
1239: (3*9)-(1+2)
123X: ((3+X)-1)*2
1244: (1+2)*(4+4)
1245: (5-(1-2))*4
1246: (2-1)*(4*6)
1247: ((1-2)+7)*4
1248: ((1-2)+4)*8
1249: (9-(1+2))*4
124X: ((1*2)*X)+4
1255: ((5*5)-2)+1
1256: ((1-2)+5)*6
1257: (1*2)*(5+7)
1258: ((5+8)-1)*2
1259: ((1+2)*5)+9
125X: ((5*X)/2)-1
1266: ((1+2)*6)+6
1267: (7-(1+2))*6
1268: (6-(1+2))*8
1269: ((1*2)*9)+6
126X: ((1+2)*X)-6
1277: ((7*7)-1)/2
1278: (1+7)+(8*2)
1279: ((9*2)-1)+7
127X: ((1*2)*7)+X
1288: ((1*2)*8)+8
1289: 9/((1+2)/8)
128X: X-((1-8)*2)
1333: (1+3)*(3+3)
1334: ((1+3)+4)*3
1335: ((1*3)+5)*3
1336: (6-(1-3))*3
1337: ((3*7)*1)+3
1338: ((1+8)*3)-3
1339: (1+3)*(9-3)
133X: ((1-3)+X)*3
1344: (4-(1-3))*4
1345: (1+3)+(4*5)
1346: 6/(1-(3/4))
1347: (4*7)-(1+3)
1348: ((1+3)*4)+8
1349: (9-(1*3))*4
134X: (X-(1+3))*4
1356: (1+5)+(6*3)
1357: (3-1)*(5+7)
1358: ((1-3)+5)*8
1359: ((1*3)*5)+9
135X: (X*3)-(1+5)
1366: ((1-3)+6)*6
1367: (7-(1*3))*6
1368: (8-(1+3))*6
1369: 6-((1-3)*9)
136X: ((1*3)*X)-6
1377: (1-7)*(3-7)
1378: (7-(1+3))*8
1379: (1+7)*(9/3)
137X: X-((1-3)*7)
1388: ((1+3)*8)-8
1389: ((8*9)/3)*1
138X: ((X-1)*8)/3
1399: (9-1)*(9/3)
139X: ((1+X)*3)-9
13XX: (1+3)+(X+X)
1444: ((1+4)*4)+4
1445: ((4*5)*1)+4
1446: ((1+6)*4)-4
1447: ((4*7)*1)-4
1448: ((1*4)*4)+8
1449: ((1-4)+9)*4
144X: (X-(1*4))*4
1455: ((5*4)-1)+5
1456: 4/(1-(5/6))
1457: ((7*4)-5)+1
1458: (1+5)*(8-4)
1459: 9-((1-4)*5)
145X: ((X-5)+1)*4
1466: ((1+4)*6)-6
1467: ((1-4)+7)*6
1468: ((1-4)+6)*8
1469: (9-(1+4))*6
146X: ((4-1)*X)-6
1477: (1+7)*(7-4)
1478: (7-(1*4))*8
1479: (1-9)*(4-7)
1488: (8-(1+4))*8
1489: 9/((4-1)/8)
149X: (1+4)+(9+X)
14XX: ((X+X)*1)+4
1555: (5-(1/5))*5
1556: (5*6)-(1+5)
1559: (1+5)*(9-5)
155X: ((X-5)*5)-1
1566: ((1*5)*6)-6
1567: ((5*6)-7)+1
1568: ((1-5)+8)*6
1569: (9-(1*5))*6
156X: (X-(1+5))*6
1578: ((1-5)+7)*8
1579: (1-7)*(5-9)
157X: (5*7)-(1+X)
1588: ((5-1)*8)-8
1589: (9-(1+5))*8
158X: (1+5)+(8+X)
1599: (1+5)+(9+9)
159X: ((9+X)*1)+5
15XX: ((X+X)-1)+5
1666: ((6-1)*6)-6
1668: 6/(1-(6/8))
1669: ((1-6)+9)*6
166X: (X-(1*6))*6
1679: (1+7)*(9-6)
167X: (1+6)+(7+X)
1688: ((1-6)+8)*8
1689: (1+6)+(8+9)
168X: (X-(1+6))*8
1699: ((9+9)*1)+6
169X: ((9+X)-1)+6
1779: (1+7)+(7+9)
177X: ((7+X)*1)+7
1788: (1+7)+(8+8)
1789: ((8+9)*1)+7
178X: ((8+X)-1)+7
1799: ((9+9)-1)+7
179X: (1-9)*(7-X)
1888: ((8+8)*1)+8
1889: ((8+9)-1)+8
188X: ((1-8)+X)*8
2223: (2+2)*(2*3)
2224: ((2+2)+2)*4
2225: ((2*5)+2)*2
2227: ((2*7)-2)*2
2228: ((2+2)+8)*2
2229: ((2+9)*2)+2
222X: (2+2)+(2*X)
2233: (2+2)*(3+3)
2234: ((2+2)+4)*3
2235: ((5*2)-2)*3
2236: ((2/2)+3)*6
2237: ((3+7)+2)*2
2238: ((3*8)-2)+2
2239: (2+2)*(9-3)
223X: ((3+X)*2)-2
2244: ((2*4)+4)*2
2245: (2+2)+(4*5)
2246: ((4*6)-2)+2
2247: (4*7)-(2+2)
2248: ((2+2)*4)+8
2249: ((4+9)*2)-2
224X: (X-(2+2))*4
2255: ((5+5)+2)*2
2256: ((5+6)*2)+2
2257: (2*5)+(7*2)
2258: ((5+8)*2)-2
2259: ((5+9)-2)*2
225X: ((5*X)-2)/2
2266: (2+6)*(6/2)
2267: ((6+7)*2)-2
2268: (8-(2+2))*6
2269: ((9*2)-6)*2
226X: ((X*2)-2)+6
2277: ((7+7)-2)*2
2278: (7-(2+2))*8
227X: ((X/2)+7)*2
2288: ((2+2)*8)-8
2289: ((9*2)-2)+8
228X: ((X-2)*2)+8
229X: X-((2-9)*2)
22XX: (2+2)+(X+X)
2333: ((2+3)+3)*3
2335: ((3*5)-3)*2
2336: (3-(2-3))*6
2337: (7-(2-3))*3
2338: (3-2)*(3*8)
2339: ((2+3)*3)+9
233X: (3*X)-(2*3)
2344: ((2+3)*4)+4
2345: (5-(2-3))*4
2346: (3-2)*(4*6)
2347: ((2-3)+7)*4
2348: ((2-3)+4)*8
2349: ((4*9)/3)*2
234X: ((X-4)+2)*3
2355: ((5*5)-3)+2
2356: ((2-3)+5)*6
2357: ((7*3)-2)+5
2358: ((3+5)*2)+8
2359: (2*3)*(9-5)
235X: ((5+X)-3)*2
2366: ((2+3)*6)-6
2367: ((6*7)/2)+3
2368: ((3*6)-2)+8
2369: (9-(2+3))*6
236X: (X-(2*3))*6
2377: (2*7)+(7+3)
2378: ((7+8)-3)*2
2379: ((7+9)/2)*3
237X: ((7*X)+2)/3
2388: (8-(2+3))*8
2389: (9-(2*3))*8
238X: (2*3)+(8+X)
2399: (2*3)+(9+9)
239X: (2+3)+(9+X)
23XX: ((X-3)*2)+X
2444: (4-(2-4))*4
2445: ((2+5)*4)-4
2446: ((4+6)*2)+4
2447: (2*4)*(7-4)
2448: (2+4)*(8-4)
2449: ((9-2)*4)-4
244X: ((4*X)/2)+4
2455: ((5+5)*2)+4
2456: (5*6)-(2+4)
2457: (4-2)*(5+7)
2458: ((5*8)/2)+4
2459: (2+4)*(9-5)
245X: (2*5)+(X+4)
2466: ((2-4)+6)*6
2467: ((7*4)-6)+2
2468: ((6+8)*2)-4
2469: 6-((2-4)*9)
246X: (X-(2+4))*6
2477: ((7+7)*2)-4
2478: ((7*8)/2)-4
2479: (2*4)+(7+9)
247X: X-((2-4)*7)
2488: 8-((2-4)*8)
2489: (9-(2+4))*8
248X: (2+4)+(8+X)
2499: (2+4)+(9+9)
249X: (2*9)+(X-4)
24XX: ((4/X)+2)*X
2557: (2*7)+(5+5)
2558: ((5/5)+2)*8
2559: 9-((2-5)*5)
255X: (5-(2/X))*5
2566: 6-((2-5)*6)
2567: ((2-5)+7)*6
2568: ((2-5)+6)*8
2569: (6/(2/5))+9
256X: ((5-2)*X)-6
2577: (2*5)+(7+7)
2578: ((2*5)-7)*8
2579: (5*7)-(2+9)
257X: (2+5)+(7+X)
2588: (8*5)-(2*8)
2589: (2+5)+(8+9)
258X: (X-(2+5))*8
259X: ((2*X)+9)-5
25XX: (2+X)*(X/5)
2666: ((6*6)/2)+6
2667: ((6*7)+6)/2
2668: ((2-6)+8)*6
2669: (2+6)*(9-6)
266X: (2+6)+(6+X)
2678: ((2-6)+7)*8
2679: (2+6)+(7+9)
267X: (2+6)*(X-7)
2688: (2+6)+(8+8)
2689: ((2*6)-9)*8
268X: ((X-8)+2)*6
2699: ((9+9)-6)*2
269X: (2-X)*(6-9)
26XX: ((X+X)-2)+6
2778: (2+7)+(7+8)
277X: ((X/7)+2)*7
2788: ((2-7)+8)*8
2789: ((9+7)*2)-8
279X: ((9+X)-2)+7
27XX: ((X+7)*2)-X
2888: ((8+8)*2)-8
2889: ((2-8)+9)*8
288X: ((8+X)-2)+8
2899: ((9+9)-2)+8
289X: ((X-9)+2)*8
28XX: ((X/X)+2)*8
29XX: (X/2)+(X+9)
3333: ((3*3)*3)-3
3334: ((3+4)*3)+3
3335: (3*3)+(3*5)
3336: (3+3)+(3*6)
3337: (3+3)*(7-3)
3338: ((3+3)-3)*8
3339: (9-(3/3))*3
333X: (3*X)-(3+3)
3344: (3*4)+(4*3)
3345: ((4+5)*3)-3
3346: ((4*6)-3)+3
3347: ((4+7)-3)*3
3348: (3+3)*(8-4)
3349: ((9-4)+3)*3
3355: (5*5)-(3/3)
3356: (5*6)-(3+3)
3357: ((3*5)-7)*3
3359: (3+3)*(9-5)
335X: (3*3)+(5+X)
3366: ((6/3)+6)*3
3367: ((7*3)-3)+6
3368: ((3*3)-6)*8
3369: (3*3)+(6+9)
336X: (X-(3+3))*6
3377: ((3/7)+3)*7
3378: (3*3)+(7+8)
3379: ((7*9)/3)+3
3388: 8/(3-(8/3))
3389: (9-(3+3))*8
338X: (3+3)+(8+X)
3399: (3+3)+(9+9)
339X: ((X*3)-9)+3
3444: ((3+4)*4)-4
3445: (5-(3-4))*4
3446: (4-3)*(4*6)
3447: ((3-4)+7)*4
3448: ((3+4)-4)*8
3449: (4*9)-(3*4)
344X: ((X-3)*4)-4
3455: ((5*5)-4)+3
3456: ((3-4)+5)*6
3457: (3*4)+(5+7)
3458: ((8-5)+3)*4
3459: ((9-5)+4)*3
345X: ((3*4)*X)/5
3466: (3*4)+(6+6)
3468: ((3*4)-8)*6
3469: ((9-6)+3)*4
346X: ((6*X)/3)+4
3477: ((7*4)-7)+3
3478: 8-((3-7)*4)
3479: (3*4)*(9-7)
347X: (3+4)+(7+X)
3489: (3+4)+(8+9)
348X: (X-(3+4))*8
3499: ((9+9)/3)*4
34XX: ((3*X)+4)-X
3556: ((5+5)*3)-6
3557: (5-3)*(5+7)
3558: ((3+5)-5)*8
3559: ((9/5)+3)*5
3566: ((3-5)+6)*6
3567: ((6+7)-5)*3
3568: 8/((5-3)/6)
3569: (3+5)*(9-6)
356X: (3+5)+(6+X)
3578: ((8-5)*7)+3
3579: (3+5)+(7+9)
357X: (3+5)*(X-7)
3588: (3+5)+(8+8)
3589: ((8-5)*9)-3
3599: (9/(3/5))+9
359X: ((X-5)*3)+9
35XX: (X-(X/5))*3
3666: 6-((3-6)*6)
3667: ((3-6)+7)*6
3668: ((3+6)-6)*8
3669: (3+6)+(6+9)
366X: ((6-3)*X)-6
3677: ((7/7)+3)*6
3678: (3+6)+(7+8)
3679: ((9-7)+6)*3
367X: (7/(3/6))+X
3688: ((8/8)+3)*6
3689: 9/((6-3)/8)
368X: ((X-8)+6)*3
3699: ((9/9)+3)*6
369X: ((9*X)/3)-6
36XX: ((X/X)+3)*6
3777: (3+7)+(7+7)
3778: ((3+7)-7)*8
3779: (9-(7/7))*3
377X: ((X-7)*7)+3
3788: ((7-3)*8)-8
3789: ((9-8)+7)*3
3799: ((9/9)+7)*3
379X: ((X-9)+7)*3
37XX: ((X+X)-3)+7
3888: ((3+8)-8)*8
3889: (3*8)*(9-8)
388X: ((8*X)-8)/3
3899: (3*8)+(9-9)
389X: ((9+X)-3)+8
38XX: (3*8)+(X-X)
3999: ((9+9)-3)+9
399X: ((9-X)+9)*3
39XX: (9-(X/X))*3
4444: (4+4)+(4*4)
4445: ((4/4)+5)*4
4446: ((4+4)-4)*6
4447: (4+4)*(7-4)
4448: (4*8)-(4+4)
4449: ((9-4)*4)+4
444X: ((4*4)-X)*4
4455: ((5+5)-4)*4
4456: (5-(4/4))*6
4457: ((7-5)+4)*4
4458: ((4+4)-5)*8
445X: ((X-5)*4)+4
4468: ((8-6)+4)*4
4469: (4+4)*(9-6)
446X: (4+4)+(6+X)
4477: (4-(4/7))*7
4478: ((8-4)*7)-4
4479: (4+4)+(7+9)
447X: (4+4)*(X-7)
4488: (4+4)+(8+8)
4489: (9*4)-(4+8)
448X: ((8*X)/4)+4
44XX: ((X*X)-4)/4
4555: ((5*5)-5)+4
4556: ((4+5)-5)*6
4557: (7-(5/5))*4
4558: (4-(5/5))*8
4559: ((9-5)*5)+4
455X: (4+5)+(5+X)
4566: ((6/6)+5)*4
4567: ((7-6)+5)*4
4568: ((4+5)-6)*8
4569: (4+5)+(6+9)
456X: ((X-6)*5)+4
4577: ((7/7)+5)*4
4578: (4+5)+(7+8)
4579: ((9-5)*7)-4
457X: ((7-5)*X)+4
4588: ((8/8)+5)*4
4589: ((9-8)+5)*4
458X: ((8/X)+4)*5
4599: ((9/9)+5)*4
459X: ((X-9)+5)*4
45XX: ((X*X)/5)+4
4666: ((4+6)-6)*6
4667: (4*6)*(7-6)
4668: (4+6)+(6+8)
4669: 6-((4-6)*9)
466X: ((6+X)/4)*6
4677: (4+6)+(7+7)
4678: ((4+6)-7)*8
4679: ((7+9)/4)*6
467X: X-((4-6)*7)
4688: 8-((4-6)*8)
4689: ((8*9)/4)+6
468X: ((X-6)*4)+8
4699: (4*6)+(9-9)
469X: (4*6)*(X-9)
46XX: (4*6)+(X-X)
4777: (7-(7/7))*4
4778: ((7-8)+7)*4
4788: ((4+7)-8)*8
4789: 9/((7-4)/8)
478X: (8/(4/7))+X
4799: (7-(9/9))*4
479X: ((9-X)+7)*4
47XX: (7-(X/X))*4
4888: ((8*8)/4)+8
4889: ((4+8)-9)*8
488X: (4+8)*(X-8)
4899: (4-(9/9))*8
489X: ((9-X)+4)*8
48XX: ((X+X)-4)+8
499X: ((9+X)-4)+9
5555: (5*5)-(5/5)
5556: (5*5)+(5-6)
5559: (5+5)+(5+9)
5566: ((5+5)-6)*6
5567: (5*5)+(6-7)
5568: (5+5)+(6+8)
5577: (5+5)+(7+7)
5578: ((5+5)-7)*8
557X: (5+7)*(X/5)
5588: (5*5)-(8/8)
5589: (5*5)+(8-9)
558X: (5-(X/5))*8
5599: (5*5)-(9/9)
559X: (5*5)+(9-X)
55XX: (5*5)-(X/X)
5666: (5-(6/6))*6
5667: (5+6)+(6+7)
5668: 6-((5-8)*6)
5669: (6*9)-(5*6)
566X: ((X-5)*6)-6
5677: (5-(7/7))*6
5678: ((7-8)+5)*6
5679: 6-((5-7)*9)
5688: ((5+6)-8)*8
5689: ((8-9)+5)*6
568X: (5*6)*(8/X)
5699: (5-(9/9))*6
569X: ((9*X)/5)+6
56XX: (5-(X/X))*6
5779: (5+7)*(9-7)
577X: X-((5-7)*7)
5788: 8-((5-7)*8)
5789: ((5+7)-9)*8
578X: (5+7)*(X-8)
579X: ((X-7)*5)+9
57XX: (X/(5/7))+X
5888: (5*8)-(8+8)
5889: 9/((8-5)/8)
588X: ((5+8)-X)*8
59XX: ((X+X)-5)+9
6666: (6+6)+(6+6)
6668: ((6+6)-8)*6
6669: ((9-6)*6)+6
666X: (6*X)-(6*6)
6679: (6+6)*(9-7)
667X: ((X-7)*6)+6
6688: 6/((8-6)/8)
6689: ((6+6)-9)*8
668X: (6+6)*(X-8)
669X: ((9-6)*X)-6
677X: ((7-X)+7)*6
6789: (6*8)/(9-7)
678X: ((6+7)-X)*8
6799: (6*7)-(9+9)
67XX: ((X-7)*X)-6
6888: 8-((6-8)*8)
6889: (8*9)-(6*8)
688X: ((8+X)/6)*8
6899: ((9+9)/6)*8
689X: ((X-8)*9)+6
699X: ((9*X)/6)+9
6XXX: ((X+X)-6)+X
779X: X-((7-9)*7)
7889: 8-((7-9)*8)
788X: (8*X)-(7*8)
789X: 9/((X-7)/8)
78XX: ((X-8)*7)+X
888X: ((X-8)*8)+8

There are a total of 566 combinations (would appreciate if JPF could confirm), 100 more than the 466 for 0..9
149 out of the 715 total combinations give no solution...

As a statistical count:
a total of 404 successful combinations are yield from 1..9
62 with 0 involved (but no 10)
162 with 10 involved (but no 0)
udosuk

Posts: 2698
Joined: 17 July 2005

udosuk wrote:Here is the list for candidate 1..10, with "X" replacing 10 There are a total of 566 combinations (would appreciate if JPF could confirm)

566 valid combinations seems to be the right number

Here is a list of solutions (not always the same as yours !)

1 1 1 8 : 8*((1+1)+1)
1 1 2 6 : 6*((2+1)+1)
1 1 2 7 : (7+1)*(2+1)
1 1 2 8 : 8*((2+1)*1)
1 1 2 9 : (9-1)*(2+1)
1 1 2 X : (10+2)*(1+1)
1 1 3 4 : 4*(3*(1+1))
1 1 3 5 : (5+1)*(3+1)
1 1 3 6 : 6*((3+1)*1)
1 1 3 7 : (7+1)*(3*1)
1 1 3 8 : 8*((3+1)-1)
1 1 3 9 : (9+3)*(1+1)
1 1 3 X : (10-(1+1))*3
1 1 4 4 : 4*((4+1)+1)
1 1 4 5 : (5*(4+1))-1
1 1 4 6 : 6*((4+1)-1)
1 1 4 7 : (7+1)*(4-1)
1 1 4 8 : (8+4)*(1+1)
1 1 4 9 : (9-1)*(4-1)
1 1 4 X : (10*(1+1))+4
1 1 5 5 : (5*5)-(1*1)
1 1 5 6 : 6*((5-1)*1)
1 1 5 7 : (7+5)*(1+1)
1 1 5 8 : 8*(5-(1+1))
1 1 6 6 : (6+6)*(1+1)
1 1 6 8 : 8*(6/(1+1))
1 1 6 9 : (9*(1+1))+6
1 1 7 X : 10+(7*(1+1))
1 1 8 8 : 8+(8*(1+1))
1 2 2 4 : 4*(2*(2+1))
1 2 2 5 : (5+1)*(2+2)
1 2 2 6 : (6+2)*(2+1)
1 2 2 7 : (7-1)*(2+2)
1 2 2 8 : 8*((2+2)-1)
1 2 2 9 : ((9+2)+1)*2
1 2 2 X : (10+2)*(2*1)
1 2 3 3 : 3*((3+1)*2)
1 2 3 4 : 4*((3+2)+1)
1 2 3 5 : (5+3)*(2+1)
1 2 3 6 : 6*((3+2)-1)
1 2 3 7 : (7*3)+(2+1)
1 2 3 8 : 8*(3*(2-1))
1 2 3 9 : (9+3)*(2*1)
1 2 3 X : ((10+3)-1)*2
1 2 4 4 : (4+4)*(2+1)
1 2 4 5 : ((5+2)-1)*4
1 2 4 6 : 6*(4*(2-1))
1 2 4 7 : ((7+4)+1)*2
1 2 4 8 : (8+4)*(2*1)
1 2 4 9 : ((9+4)-1)*2
1 2 4 X : (10-2)*(4-1)
1 2 5 5 : ((5*5)-2)+1
1 2 5 6 : 6*((5-2)+1)
1 2 5 7 : (7+5)*(2*1)
1 2 5 8 : 8*((5-2)*1)
1 2 5 9 : 9+(5*(2+1))
1 2 5 X : ((10*5)/2)-1
1 2 6 6 : 6+(6*(2+1))
1 2 6 7 : ((7+6)-1)*2
1 2 6 8 : 8*(6-(2+1))
1 2 6 9 : (9*2)+(6*1)
1 2 6 X : 10+((6+1)*2)
1 2 7 7 : ((7*7)-1)/2
1 2 7 8 : 8+((7+1)*2)
1 2 7 9 : 9+((7*2)+1)
1 2 7 X : 10+((7*2)*1)
1 2 8 8 : 8+((8*2)*1)
1 2 8 9 : 9+((8*2)-1)
1 2 8 X : 10+((8-1)*2)
1 3 3 3 : (3+3)*(3+1)
1 3 3 4 : 4*((3+3)*1)
1 3 3 5 : (5+3)*(3*1)
1 3 3 6 : ((6+3)-1)*3
1 3 3 7 : (7*3)+(3*1)
1 3 3 8 : ((8+1)*3)-3
1 3 3 9 : (9+3)*(3-1)
1 3 3 X : ((10-3)+1)*3
1 3 4 4 : (4+4)*(3*1)
1 3 4 5 : (5*4)+(3+1)
1 3 4 6 : 6/(1-(3/4))
1 3 4 7 : (7*4)-(3+1)
1 3 4 8 : 8+(4*(3+1))
1 3 4 9 : 9+((4+1)*3)
1 3 4 X : (10-4)*(3+1)
1 3 5 6 : 6+((5+1)*3)
1 3 5 7 : (7+5)*(3-1)
1 3 5 8 : 8+((5*3)+1)
1 3 5 9 : 9+((5*3)*1)
1 3 5 X : 10+((5*3)-1)
1 3 6 6 : (6+6)*(3-1)
1 3 6 7 : 7+((6*3)-1)
1 3 6 8 : 8*((6-3)*1)
1 3 6 9 : 9+((6-1)*3)
1 3 6 X : (10*3)-(6*1)
1 3 7 7 : (7-3)*(7-1)
1 3 7 8 : 8*(7-(3+1))
1 3 7 9 : 9*((7+1)/3)
1 3 7 X : 10+(7*(3-1))
1 3 8 8 : 8+(8*(3-1))
1 3 8 9 : 9*((8/3)*1)
1 3 8 X : (10-1)*(8/3)
1 3 9 9 : 9*((9-1)/3)
1 3 9 X : ((10+1)*3)-9
1 3 X X : 10+((10+3)+1)
1 4 4 4 : 4+(4*(4+1))
1 4 4 5 : (5*4)+(4*1)
1 4 4 6 : ((6+1)*4)-4
1 4 4 7 : 7+((4*4)+1)
1 4 4 8 : 8+((4*4)*1)
1 4 4 9 : 9+((4*4)-1)
1 4 4 X : (10-4)*(4*1)
1 4 5 5 : 5+((5*4)-1)
1 4 5 6 : 6/((5/4)-1)
1 4 5 7 : (7*(5-1))-4
1 4 5 8 : 8+((5-1)*4)
1 4 5 9 : 9+(5*(4-1))
1 4 5 X : ((10-5)+1)*4
1 4 6 6 : 6+(6*(4-1))
1 4 6 7 : ((7-4)+1)*6
1 4 6 8 : 8*((6-4)+1)
1 4 6 9 : (9-(4+1))*6
1 4 6 X : (10*(4-1))-6
1 4 7 7 : (7-4)*(7+1)
1 4 7 8 : 8*((7-4)*1)
1 4 7 9 : (9-1)*(7-4)
1 4 8 8 : 8*(8-(4+1))
1 4 8 9 : 9*(8/(4-1))
1 4 9 X : 10+((9+4)+1)
1 4 X X : 10+((10+4)*1)
1 5 5 5 : 5*(5-(1/5))
1 5 5 6 : (6*5)-(5+1)
1 5 5 9 : (9-5)*(5+1)
1 5 5 X : ((10-5)*5)-1
1 5 6 6 : (6*5)-(6*1)
1 5 6 7 : ((7-1)*5)-6
1 5 6 8 : ((8-5)+1)*6
1 5 6 9 : (9-5)*(6*1)
1 5 6 X : (10-6)*(5+1)
1 5 7 8 : 8*((7-5)+1)
1 5 7 9 : (9-5)*(7-1)
1 5 7 X : 10*((7/5)+1)
1 5 8 8 : 8*((8-5)*1)
1 5 8 9 : (9-(5+1))*8
1 5 8 X : 10+((8+5)+1)
1 5 9 9 : 9+((9+5)+1)
1 5 9 X : 10+((9+5)*1)
1 5 X X : 10+((10+5)-1)
1 6 6 6 : (6*(6-1))-6
1 6 6 8 : 6/(1-(6/8))
1 6 6 9 : ((9-6)+1)*6
1 6 6 X : (10-6)*(6*1)
1 6 7 9 : (9-6)*(7+1)
1 6 7 X : 10+((7+6)+1)
1 6 8 8 : 8*((8-6)+1)
1 6 8 9 : 9+((8+6)+1)
1 6 8 X : 10+((8+6)*1)
1 6 9 9 : 9+((9+6)*1)
1 6 9 X : 10+((9+6)-1)
1 7 7 9 : 9+((7+7)+1)
1 7 7 X : 10+((7+7)*1)
1 7 8 8 : 8+((8+7)+1)
1 7 8 9 : 9+((8+7)*1)
1 7 8 X : 10+((8+7)-1)
1 7 9 9 : 9+((9+7)-1)
1 7 9 X : (10-7)*(9-1)
1 8 8 8 : 8+((8+8)*1)
1 8 8 9 : 9+((8+8)-1)
1 8 8 X : ((10-8)+1)*8
2 2 2 3 : 3*((2+2)*2)
2 2 2 4 : (4+2)*(2+2)
2 2 2 5 : ((5*2)+2)*2
2 2 2 7 : ((7*2)-2)*2
2 2 2 8 : ((8+2)+2)*2
2 2 2 9 : ((9+2)*2)+2
2 2 2 X : (10*2)+(2+2)
2 2 3 3 : (3+3)*(2+2)
2 2 3 4 : ((4+2)+2)*3
2 2 3 5 : ((5*2)-2)*3
2 2 3 6 : (6+(3*2))*2
2 2 3 7 : ((7+3)+2)*2
2 2 3 8 : ((8+3)*2)+2
2 2 3 9 : (9-3)*(2+2)
2 2 3 X : ((10+3)*2)-2
2 2 4 4 : (4+(4*2))*2
2 2 4 5 : (5*4)+(2+2)
2 2 4 6 : ((6+4)+2)*2
2 2 4 7 : ((7+4)*2)+2
2 2 4 8 : 8+(4*(2+2))
2 2 4 9 : ((9+4)*2)-2
2 2 4 X : ((10+4)-2)*2
2 2 5 5 : ((5+5)+2)*2
2 2 5 6 : ((6+5)*2)+2
2 2 5 7 : (7*2)+(5*2)
2 2 5 8 : ((8+5)*2)-2
2 2 5 9 : ((9+5)-2)*2
2 2 5 X : 10+((5+2)*2)
2 2 6 6 : 6*((6+2)/2)
2 2 6 7 : ((7+6)*2)-2
2 2 6 8 : 8+((6+2)*2)
2 2 6 9 : (9+(6/2))*2
2 2 6 X : 10+((6*2)+2)
2 2 7 7 : ((7+7)-2)*2
2 2 7 8 : 8+((7*2)+2)
2 2 7 X : ((10/2)+7)*2
2 2 8 8 : (8+(8/2))*2
2 2 8 9 : (9*2)+(8-2)
2 2 8 X : 10+((8*2)-2)
2 2 9 X : 10+((9-2)*2)
2 2 X X : 10+((10+2)+2)
2 3 3 3 : (3+(3*3))*2
2 3 3 5 : ((5*3)-3)*2
2 3 3 6 : ((6+3)+3)*2
2 3 3 7 : (7-3)*(3*2)
2 3 3 8 : 8*((3+3)/2)
2 3 3 9 : 9+(3*(3+2))
2 3 3 X : (10*3)-(3*2)
2 3 4 4 : 4+(4*(3+2))
2 3 4 5 : ((5+4)+3)*2
2 3 4 6 : 6*(4*(3-2))
2 3 4 7 : (7-3)*(4+2)
2 3 4 8 : (8-4)*(3*2)
2 3 4 9 : 9*((4/3)*2)
2 3 4 X : 10+((4+3)*2)
2 3 5 5 : ((5*5)-3)+2
2 3 5 6 : 6*((5+3)/2)
2 3 5 7 : 7+((5*3)+2)
2 3 5 8 : 8+((5+3)*2)
2 3 5 9 : (9-5)*(3*2)
2 3 5 X : ((10+5)-3)*2
2 3 6 6 : 6+((6+3)*2)
2 3 6 7 : ((7*6)/2)+3
2 3 6 8 : 8+((6*3)-2)
2 3 6 9 : ((9+6)-3)*2
2 3 6 X : (10+6)*(3/2)
2 3 7 7 : 7+((7*2)+3)
2 3 7 8 : ((8+7)-3)*2
2 3 7 9 : (9+7)*(3/2)
2 3 7 X : ((10*7)+2)/3
2 3 8 8 : (8+8)*(3/2)
2 3 8 9 : (9-3)*(8/2)
2 3 8 X : 10+(8+(3*2))
2 3 9 9 : 9+(9+(3*2))
2 3 9 X : 10+((9+3)+2)
2 3 X X : 10+((10-3)*2)
2 4 4 4 : ((4+4)+4)*2
2 4 4 5 : ((5+2)*4)-4
2 4 4 6 : 6+((4*4)+2)
2 4 4 7 : (7-4)*(4*2)
2 4 4 8 : 8+((4+4)*2)
2 4 4 9 : ((9-2)*4)-4
2 4 4 X : 10+((4*4)-2)
2 4 5 5 : ((5+5)*2)+4
2 4 5 6 : 6+((5+4)*2)
2 4 5 7 : (7+5)*(4-2)
2 4 5 8 : (8-5)*(4*2)
2 4 5 9 : (9-5)*(4+2)
2 4 5 X : 10+((5*2)+4)
2 4 6 6 : (6+6)*(4-2)
2 4 6 7 : (7*(6-2))-4
2 4 6 8 : 8*(6/(4-2))
2 4 6 9 : (9-6)*(4*2)
2 4 6 X : 10+(6+(4*2))
2 4 7 7 : ((7+7)*2)-4
2 4 7 8 : ((8*7)/2)-4
2 4 7 9 : 9+(7+(4*2))
2 4 7 X : 10+(7*(4-2))
2 4 8 8 : 8+(8+(4*2))
2 4 8 9 : (9-(4+2))*8
2 4 8 X : 10+((8+4)+2)
2 4 9 9 : 9+((9+4)+2)
2 4 9 X : 10+((9*2)-4)
2 4 X X : 10*((4/10)+2)
2 5 5 7 : (7*2)+(5+5)
2 5 5 8 : 8*((5/5)+2)
2 5 5 9 : 9+(5+(5*2))
2 5 5 X : 5*(5-(2/10))
2 5 6 6 : 6+(6*(5-2))
2 5 6 7 : 7+((6*2)+5)
2 5 6 8 : 8+(6+(5*2))
2 5 6 9 : 9+((6*5)/2)
2 5 6 X : 10*((6/5)*2)
2 5 7 7 : 7+(7+(5*2))
2 5 7 8 : 8*((5*2)-7)
2 5 7 9 : (7*5)-(9+2)
2 5 7 X : 10+((7+5)+2)
2 5 8 8 : (8+(8*5))/2
2 5 8 9 : 9+((8+5)+2)
2 5 8 X : (10-(5+2))*8
2 5 9 X : (10*2)+(9-5)
2 5 X X : (10+(10/5))*2
2 6 6 6 : 6+(6+(6*2))
2 6 6 7 : ((7*6)+6)/2
2 6 6 8 : (8-6)*(6*2)
2 6 6 9 : (9-6)*(6+2)
2 6 6 X : 10+((6+6)+2)
2 6 7 8 : 8*((7-6)+2)
2 6 7 9 : 9+((7+6)+2)
2 6 7 X : (10-7)*(6+2)
2 6 8 8 : 8+((8+6)+2)
2 6 8 9 : 9*((8/6)*2)
2 6 8 X : ((10+8)-6)*2
2 6 9 9 : ((9+9)-6)*2
2 6 9 X : (10-2)*(9-6)
2 6 X X : 10+((10+6)-2)
2 7 7 8 : 8+((7+7)+2)
2 7 7 X : ((10/7)+2)*7
2 7 8 8 : 8*((8-7)+2)
2 7 8 9 : ((9+7)*2)-8
2 7 9 X : 10+((9+7)-2)
2 7 X X : (10-7)*(10-2)
2 8 8 8 : 8*((8/8)+2)
2 8 8 9 : ((9-8)+2)*8
2 8 8 X : 10+((8+8)-2)
2 8 9 9 : 9+((9+8)-2)
2 8 9 X : ((10-9)+2)*8
2 8 X X : 10+(10+(8/2))
2 9 X X : 10+((10/2)+9)
3 3 3 3 : ((3*3)*3)-3
3 3 3 4 : ((4+3)*3)+3
3 3 3 5 : (5*3)+(3*3)
3 3 3 6 : 6+((3+3)*3)
3 3 3 7 : (7+(3/3))*3
3 3 3 8 : ((8+3)-3)*3
3 3 3 9 : (9-(3/3))*3
3 3 3 X : ((10-3)*3)+3
3 3 4 4 : (4*3)+(4*3)
3 3 4 5 : ((5+4)*3)-3
3 3 4 6 : 6*((4+3)-3)
3 3 4 7 : ((7+4)-3)*3
3 3 4 8 : (8-4)*(3+3)
3 3 4 9 : 9+((4*3)+3)
3 3 5 5 : (5*5)-(3/3)
3 3 5 6 : ((6+5)-3)*3
3 3 5 7 : ((5*3)-7)*3
3 3 5 9 : (9-5)*(3+3)
3 3 5 X : 10+(5+(3*3))
3 3 6 6 : (6+(6/3))*3
3 3 6 7 : (7*(6-3))+3
3 3 6 8 : 8*((6+3)/3)
3 3 6 9 : 9+(6+(3*3))
3 3 6 X : (10-6)*(3+3)
3 3 7 7 : 7*((3/7)+3)
3 3 7 8 : 8+(7+(3*3))
3 3 7 9 : ((9*7)/3)+3
3 3 8 8 : 8/(3-(8/3))
3 3 8 9 : 9+((8-3)*3)
3 3 8 X : 10+((8+3)+3)
3 3 9 9 : 9+((9+3)+3)
3 3 9 X : ((10*3)-9)+3
3 4 4 4 : (4*(4+3))-4
3 4 4 5 : 5+((4*4)+3)
3 4 4 6 : (6-4)*(4*3)
3 4 4 7 : (7+(4/4))*3
3 4 4 8 : 8+(4+(4*3))
3 4 4 9 : (9-(4/4))*3
3 4 4 X : ((10-3)*4)-4
3 4 5 5 : ((5*5)-4)+3
3 4 5 6 : 6*((5-4)+3)
3 4 5 7 : 7+(5+(4*3))
3 4 5 8 : 8*((5+4)/3)
3 4 5 9 : ((9-5)+4)*3
3 4 5 X : (10/5)*(4*3)
3 4 6 6 : 6+(6+(4*3))
3 4 6 8 : (8-6)*(4*3)
3 4 6 9 : ((9-6)+3)*4
3 4 6 X : ((10-6)+4)*3
3 4 7 7 : (7*(7-4))+3
3 4 7 8 : 8+((7-3)*4)
3 4 7 9 : (9-7)*(4*3)
3 4 7 X : 10+((7+4)+3)
3 4 8 9 : 9+((8+4)+3)
3 4 8 X : (10+8)*(4/3)
3 4 9 9 : 9+((9-4)*3)
3 4 X X : ((10*3)-10)+4
3 5 5 6 : 6*((5/5)+3)
3 5 5 7 : (7+5)*(5-3)
3 5 5 8 : ((8+5)-5)*3
3 5 5 9 : (9-(5/5))*3
3 5 6 6 : (6+6)*(5-3)
3 5 6 7 : ((7+6)-5)*3
3 5 6 8 : 8*((6-5)*3)
3 5 6 9 : ((9-6)+5)*3
3 5 6 X : 10+((6+5)+3)
3 5 7 8 : 8+((7*3)-5)
3 5 7 9 : 9+((7+5)+3)
3 5 7 X : 10+(7*(5-3))
3 5 8 8 : 8+((8+5)+3)
3 5 8 9 : (9*(8-5))-3
3 5 9 9 : 9+((9*5)/3)
3 5 9 X : 10*((9+3)/5)
3 5 X X : (10-(10/5))*3
3 6 6 6 : 6+(6*(6-3))
3 6 6 7 : (7+(6/6))*3
3 6 6 8 : ((8+6)-6)*3
3 6 6 9 : 9+((6+6)+3)
3 6 6 X : (10*(6-3))-6
3 6 7 7 : ((7+7)-6)*3
3 6 7 8 : 8+((7+6)+3)
3 6 7 9 : ((9-7)+6)*3
3 6 7 X : 10+((7*6)/3)
3 6 8 8 : 8+((8*6)/3)
3 6 8 9 : 9*(8/(6-3))
3 6 8 X : ((10-8)+6)*3
3 6 9 9 : (9*(9-6))-3
3 6 9 X : ((10-9)+3)*6
3 6 X X : ((10/10)+3)*6
3 7 7 7 : 7+((7+7)+3)
3 7 7 8 : ((8+7)-7)*3
3 7 7 9 : (9-(7/7))*3
3 7 7 X : (10-7)+(7*3)
3 7 8 8 : 8*((8-7)*3)
3 7 8 9 : ((9-8)+7)*3
3 7 9 9 : (9+(9*7))/3
3 7 9 X : ((10-9)+7)*3
3 7 X X : 10+((10+7)-3)
3 8 8 8 : ((8+8)-8)*3
3 8 8 9 : (9-8)*(8*3)
3 8 8 X : ((10*8)-8)/3
3 8 9 9 : (9-9)+(8*3)
3 8 9 X : 10+((9+8)-3)
3 8 X X : (10-10)+(8*3)
3 9 9 9 : 9+((9+9)-3)
3 9 9 X : ((9-10)+9)*3
3 9 X X : (9-(10/10))*3
4 4 4 4 : 4+(4+(4*4))
4 4 4 5 : (5+(4/4))*4
4 4 4 6 : ((6+4)-4)*4
4 4 4 7 : (7-4)*(4+4)
4 4 4 8 : (8*4)-(4+4)
4 4 4 9 : ((9-4)*4)+4
4 4 4 X : (10*4)-(4*4)
4 4 5 5 : ((5+5)-4)*4
4 4 5 6 : 6*((5-4)*4)
4 4 5 7 : ((7-5)+4)*4
4 4 5 8 : 8+((5*4)-4)
4 4 5 X : ((10-5)*4)+4
4 4 6 8 : ((8-6)+4)*4
4 4 6 9 : (9-6)*(4+4)
4 4 6 X : 10+((6+4)+4)
4 4 7 7 : 7*(4-(4/7))
4 4 7 8 : ((8-4)*7)-4
4 4 7 9 : 9+((7+4)+4)
4 4 7 X : (10-7)*(4+4)
4 4 8 8 : 8+((8+4)+4)
4 4 8 9 : (9*4)-(8+4)
4 4 8 X : ((10-8)+4)*4
4 4 X X : ((10*10)-4)/4
4 5 5 5 : (5+(5/5))*4
4 5 5 6 : ((6+5)-5)*4
4 5 5 7 : (7-(5/5))*4
4 5 5 8 : 8*(4-(5/5))
4 5 5 9 : (9-5)+(5*4)
4 5 5 X : 10+((5+5)+4)
4 5 6 6 : 6*((6-5)*4)
4 5 6 7 : ((7-6)+5)*4
4 5 6 8 : 8*((5-6)+4)
4 5 6 9 : 9+((6+5)+4)
4 5 6 X : (10-6)+(5*4)
4 5 7 7 : ((7/7)+5)*4
4 5 7 8 : 8+((7+5)+4)
4 5 7 9 : 9+((7-4)*5)
4 5 7 X : (10*(7-5))+4
4 5 8 8 : ((8/8)+5)*4
4 5 8 9 : ((9-8)+5)*4
4 5 8 X : 10*((8+4)/5)
4 5 9 9 : ((9/9)+5)*4
4 5 9 X : ((10-9)+5)*4
4 5 X X : ((10*10)/5)+4
4 6 6 6 : ((6+6)-6)*4
4 6 6 7 : (7-6)*(6*4)
4 6 6 8 : 8+((6+6)+4)
4 6 6 9 : (9*(6-4))+6
4 6 6 X : (10+6)*(6/4)
4 6 7 7 : 7+((7+6)+4)
4 6 7 8 : (8-7)*(6*4)
4 6 7 9 : (9+7)*(6/4)
4 6 7 X : 10+(7*(6-4))
4 6 8 8 : 8+(8*(6-4))
4 6 8 9 : (9-8)*(6*4)
4 6 8 X : (10*(8-6))+4
4 6 9 9 : (9-9)+(6*4)
4 6 9 X : (10-9)*(6*4)
4 6 X X : (10-10)+(6*4)
4 7 7 7 : (7-(7/7))*4
4 7 7 8 : 8*(4-(7/7))
4 7 8 8 : 8*((7-8)+4)
4 7 8 9 : 9*(8/(7-4))
4 7 8 X : 10+((8*7)/4)
4 7 9 9 : (7-(9/9))*4
4 7 9 X : (10*(9-7))+4
4 7 X X : (7-(10/10))*4
4 8 8 8 : 8+((8*8)/4)
4 8 8 9 : ((8-9)+4)*8
4 8 8 X : (10-8)*(8+4)
4 8 9 9 : 8*(4-(9/9))
4 8 9 X : ((9-10)+4)*8
4 8 X X : 10+((10+8)-4)
4 9 9 X : 10+((9+9)-4)
5 5 5 5 : (5*5)-(5/5)
5 5 5 6 : 6*(5-(5/5))
5 5 5 9 : 9+((5+5)+5)
5 5 6 6 : (6-(6/5))*5
5 5 6 7 : (7*5)-(6+5)
5 5 6 8 : 8+((6+5)+5)
5 5 7 7 : 7+((7+5)+5)
5 5 7 8 : 8*((5-7)+5)
5 5 7 X : 10*((7+5)/5)
5 5 8 8 : (5*5)-(8/8)
5 5 8 9 : 9+((8-5)*5)
5 5 8 X : (10+5)*(8/5)
5 5 9 9 : (5*5)-(9/9)
5 5 9 X : (9-10)+(5*5)
5 5 X X : (5*5)-(10/10)
5 6 6 6 : 6*(5-(6/6))
5 6 6 7 : 7+((6+6)+5)
5 6 6 8 : ((8-5)*6)+6
5 6 6 9 : (9*6)-(6*5)
5 6 6 X : 10*((6+6)/5)
5 6 7 7 : 6*(5-(7/7))
5 6 7 8 : (8/(7-5))*6
5 6 7 9 : (9*(7-5))+6
5 6 8 8 : 8*((6-8)+5)
5 6 8 9 : (9+6)*(8/5)
5 6 8 X : (10*(8-5))-6
5 6 9 9 : 9+((9-6)*5)
5 6 9 X : ((10*9)/5)+6
5 6 X X : (10+10)*(6/5)
5 7 7 9 : (9-7)*(7+5)
5 7 7 X : 10+(7*(7-5))
5 7 8 8 : 8+(8*(7-5))
5 7 8 9 : 8*((7-9)+5)
5 7 8 X : (10-8)*(7+5)
5 7 9 X : ((10-7)*5)+9
5 7 X X : 10+((10*7)/5)
5 8 8 8 : (8*8)-(8*5)
5 8 8 9 : 9*(8/(8-5))
5 8 8 X : ((10*8)/5)+8
5 9 X X : 10+((10+9)-5)
6 6 6 6 : 6+((6+6)+6)
6 6 6 8 : (8-6)*(6+6)
6 6 6 9 : ((9-6)*6)+6
6 6 6 X : (10*6)-(6*6)
6 6 7 9 : (9-7)*(6+6)
6 6 7 X : ((10-7)*6)+6
6 6 8 8 : (8/(8-6))*6
6 6 8 9 : (9*(8-6))+6
6 6 8 X : (10-8)*(6+6)
6 6 9 X : (10*(9-6))-6
6 7 7 X : ((7-10)+7)*6
6 7 8 9 : (8/(9-7))*6
6 7 8 X : 10+((8-6)*7)
6 7 9 9 : 9*((9+7)/6)
6 7 X X : (10*(10-7))-6
6 8 8 8 : 8+(8*(8-6))
6 8 8 9 : 9*((8+8)/6)
6 8 8 X : (10+8)*(8/6)
6 8 9 9 : (9+9)*(8/6)
6 8 9 X : ((10-8)*9)+6
6 9 9 X : ((10*9)/6)+9
6 X X X : 10+((10+10)-6)
7 7 9 X : 10+((9-7)*7)
7 8 8 9 : ((9-7)*8)+8
7 8 8 X : (10*8)-(8*7)
7 8 9 X : (9/(10-7))*8
7 8 X X : 10+((10-8)*7)
8 8 8 X : ((10-8)*8)+8

JPF

PS : ready for 11,12, etc...
JPF
2017 Supporter

Posts: 4022
Joined: 06 December 2005
Location: Paris, France

Thank you udosuk and JPF for your list and confirming list, respectively. I have been using the game as part of 4 week course for advanced high school students the past 5 years and had compiled a subset of the list by hand in the process. I was always curious how many of the combinations were possible but do not have the programming accumen, nor the time needed to brute force the problem completely. Thanks for your work.

rep'nA
re'born

Posts: 551
Joined: 31 May 2007

What's the solution for 3388? It's the only one with a unique solution that I can't figure out.
PTP

Posts: 5
Joined: 02 August 2006

3388 is also the only one that the earlier version of my program couldn't figure out... Something to do with the computer thinking 8/.333333...=24.00001
udosuk

Posts: 2698
Joined: 17 July 2005

Pals, after a whole weekend's hardwork I think I've finally compiled a reliant program to work out what combinations have unique formulas...

A formula is unique if all other inequivalent formulas using the same 4 numbers do not result in the same answer (24). 2 formulas are equivalent if when all numbers are assigned as letters (variables) one could be algebraically converted to another. For example:

(7+9)+(9-1) and (9+9)-(1-7) are equivalent because (a+b)+(b-c)=(b+b)-(c-a) is always true for all a,b,c...

(4*4)+(4+4) and ((4*4)+4)+4 are equivalent because (a*a)+(a+a)=((a*a)+a)+a is always true for all a...

(3-1)*9+6 and 6-(1-3)*9 are equivalent because (a-b)*c+d=d-(b-a)*c is always true for all a,b,c,d...

(8/2)*7-4 and 7/(2/8)-4 are equivalent because (a/b)*c-d and c/(b/a)-d is always true for all a,b,c,d...

But (3+0)*(8*1) and (3-0)*(8/1) are NOT equivalent because (a+b)*(c+d) and (a-b)*(c/d) are NOT always equal...

As you shall see, combinations with 0 or 1 are very hard (0 much more so) to produce unique formulas because of the "versatility" of 0 & 1...

Anyway here are the lists of all unique combinations for 0..9 and 1..X (X=10), as well as those with 2 possible formulas:

0..9, unique formula (total 104):
1118, 1127, 1129, 1134, 1135, 1144, 1149, 1158, 1168, 1169, 1188, 1224, 1229, 1233, 1255, 1277, 1346, 1377, 1379, 1399, 1469, 1477, 1479, 1555, 1559, 1568, 1579, 1599, 1666, 1668, 1669, 1679, 1779, 1788, 1799, 1889, 2225, 2227, 2229, 2235, 2257, 2258, 2259, 2277, 2289, 2377, 2449, 2455, 2477, 2489, 2499, 2557, 2558, 2569, 2577, 2578, 2579, 2789, 2889, 3333, 3335, 3339, 3355, 3357, 3366, 3377, 3378, 3388, 3444, 3456, 3478, 3779, 3789, 4444, 4445, 4449, 4457, 4477, 4489, 4557, 4558, 4568, 4569, 4599, 4679, 4777, 4799, 4889, 4899, 5555, 5559, 5568, 5588, 5599, 5666, 5668, 5669, 5677, 5679, 5779, 6688, 6789, 6888, 7889

0..9, 2 formulas (total 106):
0699, 0789, 0888, 1126, 1157, 1166, 1225, 1227, 1228, 1245, 1247, 1279, 1333, 1336, 1338, 1356, 1357, 1378, 1444, 1446, 1449, 1455, 1456, 1457, 1458, 1459, 1466, 1467, 1556, 1567, 1578, 1688, 2223, 2234, 2237, 2244, 2249, 2255, 2266, 2267, 2269, 2333, 2335, 2345, 2389, 2447, 2457, 2479, 2559, 2566, 2588, 2678, 2689, 2699, 2778, 2899, 3334, 3337, 3344, 3347, 3368, 3445, 3447, 3455, 3459, 3469, 3556, 3557, 3559, 3566, 3567, 3588, 3589, 3599, 3677, 3777, 4447, 4468, 4469, 4478, 4479, 4555, 4559, 4567, 4577, 4588, 4589, 4778, 5556, 5567, 5578, 5589, 5688, 5689, 5699, 5788, 5789, 5888, 5889, 6666, 6668, 6669, 6679, 6689, 6889, 6899

1..X, unique formula (total 166):
1118, 1127, 1129, 1134, 1135, 113X, 1144, 1149, 114X, 1158, 1168, 1169, 117X, 1188, 1224, 1229, 1233, 1255, 1277, 128X, 1346, 1377, 1379, 138X, 1399, 139X, 13XX, 1469, 146X, 1477, 1479, 149X, 1555, 1559, 155X, 1568, 1579, 1599, 15XX, 1666, 1668, 1669, 1679, 1779, 1788, 1799, 179X, 1889, 188X, 2225, 2227, 2229, 2235, 223X, 2257, 2258, 2259, 2277, 227X, 2289, 229X, 2377, 23XX, 2449, 2455, 245X, 2477, 2489, 2499, 24XX, 2557, 2558, 255X, 2569, 2577, 2578, 2579, 259X, 277X, 2789, 2889, 29XX, 3333, 3335, 3339, 3355, 3357, 3366, 3377, 3378, 3388, 339X, 3444, 344X, 3456, 3478, 34XX, 35XX, 366X, 367X, 368X, 3779, 3789, 388X, 399X, 39XX, 4444, 4445, 4449, 4457, 4477, 447X, 4489, 44XX, 4557, 4558, 455X, 4568, 4569, 457X, 4599, 466X, 4679, 4777, 478X, 4799, 47XX, 4889, 4899, 489X, 499X, 5555, 5559, 5568, 557X, 5588, 5599, 559X, 55XX, 5666, 5668, 5669, 5677, 5679, 5779, 577X, 578X, 579X, 57XX, 59XX, 666X, 667X, 6688, 677X, 6789, 67XX, 6888, 689X, 699X, 6XXX, 779X, 7889, 788X, 789X, 78XX, 888X

1..X, 2 formulas (total 148):
1126, 112X, 1157, 1166, 1225, 1227, 1228, 1245, 1247, 125X, 1279, 1333, 1336, 1338, 133X, 1356, 1357, 135X, 1378, 137X, 1444, 1446, 1449, 1455, 1456, 1457, 1458, 1459, 145X, 1466, 1467, 1556, 1567, 156X, 1578, 157X, 158X, 1688, 169X, 2223, 222X, 2234, 2237, 2244, 2249, 2255, 2266, 2267, 2269, 22XX, 2333, 2335, 233X, 2345, 235X, 237X, 2389, 238X, 2447, 2457, 2479, 249X, 2559, 2566, 2588, 258X, 25XX, 2678, 267X, 2689, 2699, 269X, 26XX, 2778, 27XX, 288X, 2899, 289X, 3334, 3337, 333X, 3344, 3347, 3368, 338X, 3445, 3447, 3455, 3459, 345X, 3469, 3556, 3557, 3559, 3566, 3567, 356X, 3588, 3589, 3599, 359X, 3677, 36XX, 3777, 377X, 37XX, 4447, 444X, 445X, 4468, 4469, 446X, 4478, 4479, 4555, 4559, 4567, 4577, 4588, 4589, 459X, 45XX, 468X, 4778, 5556, 5567, 5578, 5589, 558X, 5688, 5689, 5699, 569X, 56XX, 5788, 5789, 5888, 5889, 588X, 6666, 6668, 6669, 6679, 6689, 668X, 669X, 6889, 6899

And here are the counts of #formulas vs #combinations:
Code: Select all
`For 0..9: 0    249 1    104 2    106 3     60 4     52 5     21 6     49 7     23 8     13 9      710      211      112     1614      215      116      118      327      230      144      2total=715For 1..X: 0    149 1    166 2    148 3     77 4     66 5     25 6     23 7     26 8     16 9      810      211      412      114      215      116      1total=715`

New challenges :

1. Find the sole combination from 0..9 that results in 11 different formulas...

2. The only 2 combinations that result in 10 different formulas are both from 2..9, could you find them?

3. There is only 1 combination with 12 formulas that is without a 0, find it...

4. Are the 15er and 16er from 0..9 & 1..X respectively same as each other?

Also, back to the basics, the following unique combinations with X are pretty tricky (not necessary fractional):
Code: Select all
`139X24XX255X277X388X44XX`

Anyone who's interested about the programming aspect could request me for the program code...

Here are the answers to the new challenges in tiny text (copy & paste the tiny text to Notepad to view):

1. 1699

2. 2346, 2448

3. 1266

4. The 15er is the same (1789), the 16ers are different (0226 & 168X respectively).

5. ((10+1)*3)-9, ((4/10)+2)*10, (5-(2/10))*5, ((10/7)+2)*7, ((10*8)-8)/3, ((10*10)-4)/4

udosuk

Posts: 2698
Joined: 17 July 2005

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