The Gauntlet

What this is

"The Gauntlet" is proposed primarily as a tournament structure to be used in early season tournaments that might have teams of a variety of different levels; it has the following features:

You will think it unusual that the top two play in the first round. The point is that I stuck mostly weaker competition as far from the top team as I could, and I don't want the second seed then to be playing all that weaker competition and cakewalking into the last round. This is something of a judgment call, and, indeed, is a reason not to really use this as a "championship" as such; this tournament is designed so that each team plays teams close to its level.
A miracle occurs: if team A wins more games than team B, team A will end with higher seed than team B. (Try it.) Further, the scores will follow a binomial distribution; for a four round event, there will be one team each with four wins and four losses, four each with three wins and three losses, and six with two of each. In particular, any team can make it to number 1 by winning all four games.
An interesting observation: in each round, assuming again that it's seeded correctly, the best team that wins beats the best team that loses that round, the second best team that wins in that round beats the second best team that loses in that round, etc.

The kicker

If there are no upsets, each team beats teams that end with one fewer win than it ends up with. (For example, a team that ends a four round tournament with 3 wins will have beat 3 teams with 2 wins each and lost to the team that won 4.) This is possible because, for example, there are four teams with one loss each, so that, together, they lose a total of four games, the number of games won by the team with four wins. (Similarly for other numbers of wins; the number of teams won by each tier of teams is equal to the number lost by the next tier down.) This kicker is, at some level, the point; I think early season tournaments should, to whatever extent possible, leave a team with a number of wins that is proportionate to the strength of the team, and should ideally give the team a chance to play most of its games against teams of similar strength. (The mathematical coincidence is, if nothing else, a solution in search of a problem.)
Another way of thinking of this: if there are no upsets, any two teams that play each other have the same record outside of that game. For example, in the third round of the 8 round tournament, the 5 seed plays the 15 seed; each team goes 6-1 in the other seven games.

How to read this

Seed the teams, the best team is 1, the next is 2, and so on; now let them play against each other as indicated, but if the lower seed wins, they switch places. (Just like you did at summer camp at the air hockey table; if you beat number 1, you are the new number 1... for the time being.) It works out that, even with this switching, no two teams can play each other twice.

1-2	3-4
1-3	2-4

1-2	3-5	4-6	7-8
1-3	2-5	4-7	6-8
1-4	2-6	3-7	5-8

1-2	3-6	4-7	5-9	8-12	10-13	11-14	15-16
1-3	2-6	4-8	5-10	7-12	9-13	11-15	14-16
1-4	2-7	3-8	5-11	6-12	9-14	10-15	13-16
1-5	2-9	3-10	4-11	6-13	7-14	8-15	12-16

1-2	3-7	4-8	5-10	6-13	9-17	11-18	12-19	14-21	15-22	16-24	20-27	23-28	25-29	26-30	31-32
1-3	2-7	4-9	5-11	6-14	8-17	10-18	12-20	13-21	15-23	16-25	19-27	22-28	24-29	26-31	30-32
1-4	2-8	3-9	5-12	6-15	7-17	10-19	11-20	13-22	14-23	16-26	18-27	21-28	24-30	25-31	29-32
1-5	2-10	3-11	4-12	6-16	7-18	8-19	9-20	13-24	14-25	15-26	17-27	21-29	22-30	23-31	28-32
1-6	2-13	3-14	4-15	5-16	7-21	8-22	9-23	10-24	11-25	12-26	17-28	18-29	19-30	20-31	27-32

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

1-2	3-9	4-10	5-12	6-15	7-19	8-24	11-30	13-31	14-32	16-34	17-35	18-37	20-40	21-41	22-43	23-46	25-50	26-51	27-53	28-56	29-60	33-65	36-66	38-67	39-68	42-70	44-71	45-72	47-74	48-75	49-77	52-80	54-81	55-82	57-84	58-85	59-87	61-90	62-91	63-93	64-96	69-100	73-101	76-102	78-103	79-104	83-106	86-107	88-108	89-109	92-111	94-112	95-113	97-115	98-116	99-118	105-121	110-122	114-123	117-124	119-125	120-126	127-128
1-3	2-9	4-11	5-13	6-16	7-20	8-25	10-30	12-31	14-33	15-34	17-36	18-38	19-40	21-42	22-44	23-47	24-50	26-52	27-54	28-57	29-61	32-65	35-66	37-67	39-69	41-70	43-71	45-73	46-74	48-76	49-78	51-80	53-81	55-83	56-84	58-86	59-88	60-90	62-92	63-94	64-97	68-100	72-101	75-102	77-103	79-105	82-106	85-107	87-108	89-110	91-111	93-112	95-114	96-115	98-117	99-119	104-121	109-122	113-123	116-124	118-125	120-127	126-128
1-4	2-10	3-11	5-14	6-17	7-21	8-26	9-30	12-32	13-33	15-35	16-36	18-39	19-41	20-42	22-45	23-48	24-51	25-52	27-55	28-58	29-62	31-65	34-66	37-68	38-69	40-70	43-72	44-73	46-75	47-76	49-79	50-80	53-82	54-83	56-85	57-86	59-89	60-91	61-92	63-95	64-98	67-100	71-101	74-102	77-104	78-105	81-106	84-107	87-109	88-110	90-111	93-113	94-114	96-116	97-117	99-120	103-121	108-122	112-123	115-124	118-126	119-127	125-128
1-5	2-12	3-13	4-14	6-18	7-22	8-27	9-31	10-32	11-33	15-37	16-38	17-39	19-43	20-44	21-45	23-49	24-53	25-54	26-55	28-59	29-63	30-65	34-67	35-68	36-69	40-71	41-72	42-73	46-77	47-78	48-79	50-81	51-82	52-83	56-87	57-88	58-89	60-93	61-94	62-95	64-99	66-100	70-101	74-103	75-104	76-105	80-106	84-108	85-109	86-110	90-112	91-113	92-114	96-118	97-119	98-120	102-121	107-122	111-123	115-125	116-126	117-127	124-128
1-6	2-15	3-16	4-17	5-18	7-23	8-28	9-34	10-35	11-36	12-37	13-38	14-39	19-46	20-47	21-48	22-49	24-56	25-57	26-58	27-59	29-64	30-66	31-67	32-68	33-69	40-74	41-75	42-76	43-77	44-78	45-79	50-84	51-85	52-86	53-87	54-88	55-89	60-96	61-97	62-98	63-99	65-100	70-102	71-103	72-104	73-105	80-107	81-108	82-109	83-110	90-115	91-116	92-117	93-118	94-119	95-120	101-121	106-122	111-124	112-125	113-126	114-127	123-128
1-7	2-19	3-20	4-21	5-22	6-23	8-29	9-40	10-41	11-42	12-43	13-44	14-45	15-46	16-47	17-48	18-49	24-60	25-61	26-62	27-63	28-64	30-70	31-71	32-72	33-73	34-74	35-75	36-76	37-77	38-78	39-79	50-90	51-91	52-92	53-93	54-94	55-95	56-96	57-97	58-98	59-99	65-101	66-102	67-103	68-104	69-105	80-111	81-112	82-113	83-114	84-115	85-116	86-117	87-118	88-119	89-120	100-121	106-123	107-124	108-125	109-126	110-127	122-128
1-8	2-24	3-25	4-26	5-27	6-28	7-29	9-50	10-51	11-52	12-53	13-54	14-55	15-56	16-57	17-58	18-59	19-60	20-61	21-62	22-63	23-64	30-80	31-81	32-82	33-83	34-84	35-85	36-86	37-87	38-88	39-89	40-90	41-91	42-92	43-93	44-94	45-95	46-96	47-97	48-98	49-99	65-106	66-107	67-108	68-109	69-110	70-111	71-112	72-113	73-114	74-115	75-116	76-117	77-118	78-119	79-120	100-122	101-123	102-124	103-125	104-126	105-127	121-128

1-2	3-10	4-11	5-13	6-16	7-20	8-25	9-31	12-38	14-39	15-40	17-42	18-43	19-45	21-48	22-49	23-51	24-54	26-58	27-59	28-61	29-64	30-68	32-73	33-74	34-76	35-79	36-83	37-88	41-94	44-95	46-96	47-97	50-99	52-100	53-101	55-103	56-104	57-106	60-109	62-110	63-111	65-113	66-114	67-116	69-119	70-120	71-122	72-125	75-129	77-130	78-131	80-133	81-134	82-136	84-139	85-140	86-142	87-145	89-149	90-150	91-152	92-155	93-159	98-164	102-165	105-166	107-167	108-168	112-170	115-171	117-172	118-173	121-175	123-176	124-177	126-179	127-180	128-182	132-185	135-186	137-187	138-188	141-190	143-191	144-192	146-194	147-195	148-197	151-200	153-201	154-202	156-204	157-205	158-207	160-210	161-211	162-213	163-216	169-220	174-221	178-222	181-223	183-224	184-225	189-227	193-228	196-229	198-230	199-231	203-233	206-234	208-235	209-236	212-238	214-239	215-240	217-242	218-243	219-245	226-248	232-249	237-250	241-251	244-252	246-253	247-254	255-256
1-3	2-10	4-12	5-14	6-17	7-21	8-26	9-32	11-38	13-39	15-41	16-42	18-44	19-46	20-48	22-50	23-52	24-55	25-58	27-60	28-62	29-65	30-69	31-73	33-75	34-77	35-80	36-84	37-89	40-94	43-95	45-96	47-98	49-99	51-100	53-102	54-103	56-105	57-107	59-109	61-110	63-112	64-113	66-115	67-117	68-119	70-121	71-123	72-126	74-129	76-130	78-132	79-133	81-135	82-137	83-139	85-141	86-143	87-146	88-149	90-151	91-153	92-156	93-160	97-164	101-165	104-166	106-167	108-169	111-170	114-171	116-172	118-174	120-175	122-176	124-178	125-179	127-181	128-183	131-185	134-186	136-187	138-189	140-190	142-191	144-193	145-194	147-196	148-198	150-200	152-201	154-203	155-204	157-206	158-208	159-210	161-212	162-214	163-217	168-220	173-221	177-222	180-223	182-224	184-226	188-227	192-228	195-229	197-230	199-232	202-233	205-234	207-235	209-237	211-238	213-239	215-241	216-242	218-244	219-246	225-248	231-249	236-250	240-251	243-252	245-253	247-255	254-256
1-4	2-11	3-12	5-15	6-18	7-22	8-27	9-33	10-38	13-40	14-41	16-43	17-44	19-47	20-49	21-50	23-53	24-56	25-59	26-60	28-63	29-66	30-70	31-74	32-75	34-78	35-81	36-85	37-90	39-94	42-95	45-97	46-98	48-99	51-101	52-102	54-104	55-105	57-108	58-109	61-111	62-112	64-114	65-115	67-118	68-120	69-121	71-124	72-127	73-129	76-131	77-132	79-134	80-135	82-138	83-140	84-141	86-144	87-147	88-150	89-151	91-154	92-157	93-161	96-164	100-165	103-166	106-168	107-169	110-170	113-171	116-173	117-174	119-175	122-177	123-178	125-180	126-181	128-184	130-185	133-186	136-188	137-189	139-190	142-192	143-193	145-195	146-196	148-199	149-200	152-202	153-203	155-205	156-206	158-209	159-211	160-212	162-215	163-218	167-220	172-221	176-222	179-223	182-225	183-226	187-227	191-228	194-229	197-231	198-232	201-233	204-234	207-236	208-237	210-238	213-240	214-241	216-243	217-244	219-247	224-248	230-249	235-250	239-251	242-252	245-254	246-255	253-256
1-5	2-13	3-14	4-15	6-19	7-23	8-28	9-34	10-39	11-40	12-41	16-45	17-46	18-47	20-51	21-52	22-53	24-57	25-61	26-62	27-63	29-67	30-71	31-76	32-77	33-78	35-82	36-86	37-91	38-94	42-96	43-97	44-98	48-100	49-101	50-102	54-106	55-107	56-108	58-110	59-111	60-112	64-116	65-117	66-118	68-122	69-123	70-124	72-128	73-130	74-131	75-132	79-136	80-137	81-138	83-142	84-143	85-144	87-148	88-152	89-153	90-154	92-158	93-162	95-164	99-165	103-167	104-168	105-169	109-170	113-172	114-173	115-174	119-176	120-177	121-178	125-182	126-183	127-184	129-185	133-187	134-188	135-189	139-191	140-192	141-193	145-197	146-198	147-199	149-201	150-202	151-203	155-207	156-208	157-209	159-213	160-214	161-215	163-219	166-220	171-221	175-222	179-224	180-225	181-226	186-227	190-228	194-230	195-231	196-232	200-233	204-235	205-236	206-237	210-239	211-240	212-241	216-245	217-246	218-247	223-248	229-249	234-250	238-251	242-253	243-254	244-255	252-256
1-6	2-16	3-17	4-18	5-19	7-24	8-29	9-35	10-42	11-43	12-44	13-45	14-46	15-47	20-54	21-55	22-56	23-57	25-64	26-65	27-66	28-67	30-72	31-79	32-80	33-81	34-82	36-87	37-92	38-95	39-96	40-97	41-98	48-103	49-104	50-105	51-106	52-107	53-108	58-113	59-114	60-115	61-116	62-117	63-118	68-125	69-126	70-127	71-128	73-133	74-134	75-135	76-136	77-137	78-138	83-145	84-146	85-147	86-148	88-155	89-156	90-157	91-158	93-163	94-164	99-166	100-167	101-168	102-169	109-171	110-172	111-173	112-174	119-179	120-180	121-181	122-182	123-183	124-184	129-186	130-187	131-188	132-189	139-194	140-195	141-196	142-197	143-198	144-199	149-204	150-205	151-206	152-207	153-208	154-209	159-216	160-217	161-218	162-219	165-220	170-221	175-223	176-224	177-225	178-226	185-227	190-229	191-230	192-231	193-232	200-234	201-235	202-236	203-237	210-242	211-243	212-244	213-245	214-246	215-247	222-248	228-249	233-250	238-252	239-253	240-254	241-255	251-256
1-7	2-20	3-21	4-22	5-23	6-24	8-30	9-36	10-48	11-49	12-50	13-51	14-52	15-53	16-54	17-55	18-56	19-57	25-68	26-69	27-70	28-71	29-72	31-83	32-84	33-85	34-86	35-87	37-93	38-99	39-100	40-101	41-102	42-103	43-104	44-105	45-106	46-107	47-108	58-119	59-120	60-121	61-122	62-123	63-124	64-125	65-126	66-127	67-128	73-139	74-140	75-141	76-142	77-143	78-144	79-145	80-146	81-147	82-148	88-159	89-160	90-161	91-162	92-163	94-165	95-166	96-167	97-168	98-169	109-175	110-176	111-177	112-178	113-179	114-180	115-181	116-182	117-183	118-184	129-190	130-191	131-192	132-193	133-194	134-195	135-196	136-197	137-198	138-199	149-210	150-211	151-212	152-213	153-214	154-215	155-216	156-217	157-218	158-219	164-220	170-222	171-223	172-224	173-225	174-226	185-228	186-229	187-230	188-231	189-232	200-238	201-239	202-240	203-241	204-242	205-243	206-244	207-245	208-246	209-247	221-248	227-249	233-251	234-252	235-253	236-254	237-255	250-256
1-8	2-25	3-26	4-27	5-28	6-29	7-30	9-37	10-58	11-59	12-60	13-61	14-62	15-63	16-64	17-65	18-66	19-67	20-68	21-69	22-70	23-71	24-72	31-88	32-89	33-90	34-91	35-92	36-93	38-109	39-110	40-111	41-112	42-113	43-114	44-115	45-116	46-117	47-118	48-119	49-120	50-121	51-122	52-123	53-124	54-125	55-126	56-127	57-128	73-149	74-150	75-151	76-152	77-153	78-154	79-155	80-156	81-157	82-158	83-159	84-160	85-161	86-162	87-163	94-170	95-171	96-172	97-173	98-174	99-175	100-176	101-177	102-178	103-179	104-180	105-181	106-182	107-183	108-184	129-200	130-201	131-202	132-203	133-204	134-205	135-206	136-207	137-208	138-209	139-210	140-211	141-212	142-213	143-214	144-215	145-216	146-217	147-218	148-219	164-221	165-222	166-223	167-224	168-225	169-226	185-233	186-234	187-235	188-236	189-237	190-238	191-239	192-240	193-241	194-242	195-243	196-244	197-245	198-246	199-247	220-248	227-250	228-251	229-252	230-253	231-254	232-255	249-256
1-9	2-31	3-32	4-33	5-34	6-35	7-36	8-37	10-73	11-74	12-75	13-76	14-77	15-78	16-79	17-80	18-81	19-82	20-83	21-84	22-85	23-86	24-87	25-88	26-89	27-90	28-91	29-92	30-93	38-129	39-130	40-131	41-132	42-133	43-134	44-135	45-136	46-137	47-138	48-139	49-140	50-141	51-142	52-143	53-144	54-145	55-146	56-147	57-148	58-149	59-150	60-151	61-152	62-153	63-154	64-155	65-156	66-157	67-158	68-159	69-160	70-161	71-162	72-163	94-185	95-186	96-187	97-188	98-189	99-190	100-191	101-192	102-193	103-194	104-195	105-196	106-197	107-198	108-199	109-200	110-201	111-202	112-203	113-204	114-205	115-206	116-207	117-208	118-209	119-210	120-211	121-212	122-213	123-214	124-215	125-216	126-217	127-218	128-219	164-227	165-228	166-229	167-230	168-231	169-232	170-233	171-234	172-235	173-236	174-237	175-238	176-239	177-240	178-241	179-242	180-243	181-244	182-245	183-246	184-247	220-249	221-250	222-251	223-252	224-253	225-254	226-255	248-256