Here is one solution to the logic problem The Camp of Israel in the May 2007 issue of Minutes With Messiah.
The three tribes with the lowest populations (32,200; 35,400; 40,500) camped on the west (clue 3). The prince with the lowest population does not share an initial with another prince (clue 3). He is not Pagiel (clue 9) or Shelumiel (clue 12), so he must be Gamaliel. (Or you could have just read clue 8.) He is not the prince of Dan (clue 6), Naphtali or Zebulun (clue 7), Reuben (clue 8), Benjamin or Issachar (clue 11), Gad (clue 12), Judah (clue 1), Simeon (whose prince is Shelumiel) (clue 8), Ephraim (clue 9), or Asher (clue 5). Gamaliel must be the prince of the tribe of Manasseh.
Abidan also camped on the west (clue 10). He was prince of Benjamin (clues 11, 10), and his tribe had a population of 35,400. The other prince to the west was not Nashon (clue 14), Eliab (clue 1), Elizur or Shelumiel (clue 12), Pagiel (clue 2), Eliasaph (clue 4), Nethanael (clues 5, 14), Ahiezer or Ahira (clue 5). He must be Elishama and his tribe must have a population of 40,500. His tribe cannot be Dan, Naphtali, Zebulun, Simeon, Reuben, Issachar, Gad, Judah, or Asher (same clues as for Gamaliel, above). He must be prince of the tribe of Ephraim. Those are the camps on the west.
Pagiel is the prince whose tribe has a population of 41,500 (clue 9). Eliasaph’s tribe has a population of 45,650 (clue 9).
The tribe of Judah, Eliab’s tribe, and the tribe with a population of 54,400 all camped on the east (clue 1). So did Issachar (clue 4). Judah’s prince is not Pagiel or Shelumiel (clue 2), Eliasaph (clue 4), Ahiezer or Ahira (clue 10), or Elizur (clue 12). The princes of Judah and Issachar must be Nashon and Nethanael. Nashon cannot be prince of Issachar, population 54,400 (clue 11), so he must be prince of Judah, whose population must be 74,600 (clue 14). Eliab was prince of Zebulun (clue 13). His tribe’s population was not 53,400 (clue 4), 45,650 or 46,500 (clue 5), or 62,700 (clue 13). The population of the tribe of Zebulon must be either 57,400 or 59,300.
The tribes of Reuben and Simeon, and the tribe with a population of 45,650 camped either on the north or south (clue 8). Shelumiel’s tribe (Simeon) camped on the south (clue 12), so Reuben and Gad also camped on the south (clue 12). Asher, Dan, and Naphtali are the tribes camped on the north. Gad had a population of 45,650, so that tribe’s prince is Eliasaph and Reuben’s prince is Elizur. Pagiel is prince of Asher (by elimination).
The tribes with populations in the fifty-thousands were Issachar, Naphtali, Zebulun, and either Reuben or Simeon (clue 7). Neither Ahiezer nor Ahira were princes with a population of 46,500. One was prince of Dan and the other of Naphtali (by elimination). So the tribe of Dan has a population of 62,700, and Simeon a population of 59,300. Zebulun has a population of 57,400 (clue 13). Ahira is prince of Naphtali (clue 13) and Ahiezer is prince of the tribe of Dan.
See Numbers 2 for a complete summary of tribes, princes, and populations to each direction of the Tabernacle.