## Description

1 Finding particular sequences of prime numbers

Write a program that ﬁnds all sequences of 6 consecutive prime 5-digit numbers, so of the form (a,b,c,d,e,f) with b = a+2, c = b+4, d = c+6, e = d+8, and f = e+10. So a, b, c, d and e are all 5-digit prime numbers and no number between a and b, between b and c, between c and d, between d and e, and between e and f is prime. The expected output is:

The solutions are:

13901 13903 13907 13913 13921 13931 21557 21559 21563 21569 21577 21587 28277 28279 28283 28289 28297 28307 55661 55663 55667 55673 55681 55691 68897 68899 68903 68909 68917 68927

2 R Decoding a multiplication

Write a program that decodes all multiplications of the form

* * * x * * ———* * * * * * * ———* * * *

such that the sum of all digits in all 4 columns is constant. The expected output is:

411 * 13 = 5343, all columns adding up to 10. 425 * 23 = 9775, all columns adding up to 18.

1

3 Decoding a sequence of operations

Write a program that ﬁnds all possible ways of inserting + and – signs in the sequence 123456789 (at most one sign before any digit) such that the resulting arithmetic expression evaluates to 100. Here are a few hints.

• 1 can either be preceded by -, or optionally be preceded by +; so 1 starts a negative or a positive number. • Allotherdigitscanbeprecededby – andstartanewnumbertobesubtractedtotherunning sum, or be preceded by + and start a new number to be added to the running sum, or not be preceded by any sign and be part of a number which it is not the leftmost digit of. That gives 38 possibilities for all digits from 2 to 9. We can generate a number N in [0,38 −1]. Then we can: – consider the remainder division of N by 3 to decide which of the three possibilities applies to 2; – consider the remainder division of N 3 by 3 to decide which of the three possibilities applies to 3; – consider the remainder division of N 32 by 3 to decide which of the three possibilities applies to 4; – …

The expected output is (the ordering could be diﬀerent):

1 + 23 – 4 + 5 + 6 + 78 – 9 = 100 123 – 4 – 5 – 6 – 7 + 8 – 9 = 100 123 + 45 – 67 + 8 – 9 = 100 123 + 4 – 5 + 67 – 89 = 100 12 + 3 + 4 + 5 – 6 – 7 + 89 = 100 123 – 45 – 67 + 89 = 100 12 – 3 – 4 + 5 – 6 + 7 + 89 = 100 1 + 2 + 34 – 5 + 67 – 8 + 9 = 100 1 + 2 + 3 – 4 + 5 + 6 + 78 + 9 = 100 -1 + 2 – 3 + 4 + 5 + 6 + 78 + 9 = 100 12 + 3 – 4 + 5 + 67 + 8 + 9 = 100 1 + 23 – 4 + 56 + 7 + 8 + 9 = 100