1, 10465
2, 10
3, 1
4, 30
5, 0.93
6, 2
7, First, consider the possibilities he offers:
- 1119- 1134- 1138- 1106- 1124- 1107- 1126- 1118- 1105- 1135Next, it is known that Anh told Binh and Minh about 2 random numbers and the position of the digits in the password, which they know is different. This implies that the two digits revealed are different and occupy different places in the password.Let's go through the conversation:Binh said he didn't know the password at first, but he's sure Minh doesn't know it either. This implies that the two digits revealed did not lead Binh to a unique password. If the revealed digits are unique, Binh can infer a password based on them. Therefore, the digits revealed to Binh cannot be unique.Minh admits he didn't know the password at first, but now knows after hearing the two digits and their location. This means that the digits revealed to Minh result in a unique password.Later, Binh realized that he also knew the password after listening to Minh say it. This suggests that the digits revealed to Minh were not included in the other possible passwords that Binh had initially considered.Based on the given dialogue and inference, we can determine the password of the mysterious box.From the list of possibilities, consider the removal process:- If the revealed digits are 1 and 1 in positions 1 and 3, it will not result in a unique password. For example, both 1119 and 1134 share this pattern.- If the revealed digits are 1 and 1 in positions 1 and 4, it will not result in a unique password. For example, both 1119 and 1118 share this pattern.- If the revealed digits are 1 and 1 in positions 1 and 2, it will not result in a unique password. For example, both 1119 and 1105 share this pattern.- If the revealed digits are 1 and 1 in positions 2 and 3, it will not result in a unique password. For example, both 1134 and 1138 share this pattern.- If the revealed digits are 1 and 1 in positions 2 and 4, it will not result in a unique password. For example, both 1124 and 1107 share this pattern.Based on the elimination process, we are left with only one possibility: 1106. In this case the revealed digits will be 1 and 1 in positions 3 and 4. This pattern is unique for password 1106. and both Binh and Minh infer the correct password based on the given information.Therefore, the password for the mystery box is 1106.8, To find x, y, and z in the given sequences, let's analyze each sequence separately:
(1): 4 - 27 - 90 - 214 - 428 - 761 - XWe can see that each term is obtained by multiplying the previous term by a constant and then adding a value. Let's see the pattern:2^2 - 1 = 43^3 - 2 = 274^2 - 2 = 905^3 - 5 = 2146^2 - 6 = 4287^3 - 5 = 761Therefore, the next term would be 8^2 - 8 = 120.(2): 1 - 16 - 81 - 256 - 625 - 1296 - YThis sequence consists of perfect cubes. Let's see the pattern:1^4 = 12^4 = 163^4 = 814^4 = 2565^4 = 6256^4 = 1296Therefore, the next term would be 7^4 = 2401.(3): 2 - 5 - 18 - 65 - 218 - 693 - ZIn this sequence, each term is obtained by multiplying the previous term by 3 and then subtracting a power of 2 minus 1. Let's see the pattern:2x3-1 = 55x3^2-1 = 1818x3^3-1 = 6565x3^4-1 = 218218x3^5-1 = 693Therefore, the next term would be 693x3^6-1 = 2074.Now, let's determine which sequence will have the biggest number when searching for the 10th term.For sequence (1), we have:4, 27, 90, 214, 428, 761, 120, ?, ?, ?For sequence (2), we have:1, 16, 81, 256, 625, 1296, 2401, ?, ?, ?For sequence (3), we have:2, 5, 18, 65, 218, 693, 2074, ?, ?, ?To find the 10th term, we need to apply the pattern for each sequence until we reach the 10th term.For sequence (1), using the pattern, we can find that the 10th term would be 10^2 - 10 = 90.For sequence (2), the 10th term would be 10^4 = 10000.For sequence (3), the 10th term would be 693x3^9-1 = 1487830.Comparing the 10th terms, we see that the sequence with the biggest number is (3).